Special functions associated with automorphisms of the space of solutions to special double confluent Heun equation
aa r X i v : . [ m a t h . C V ] M a r SPECIAL FUNCTIONS ASSOCIATED WITHAUTOMORPHISMS OF THE SPACE OF SOLUTIONS TOSPECIAL DOUBLE CONFLUENT HEUN EQUATION
S.I. TERTYCHNIY
Abstract.
The family of quads of interrelated functions holomorphic on theuniversal cover of the complex plane without zero (for brevity, pqrs -functions),revealing a number of remarkable properties, is introduced. In particular, un-der certain conditions the transformations of the argument z of pqrs -functionsrepresented by lifts of the replacements z Ø ´ { z , z Ø ´ z , and z Ø { z prove to be equivalent to linear transformations with known coefficients. Pqrs -functions are utilized for constructing of linear operators acting as automor-phisms on the space of solutions to the special double confluent Heun equation(sDCHE). Earlier such symmetries were known to exist only in the case ofinteger value of one of the constant parameters. Then pqrs -functions appear aspolynomials, a constructive algorithm of their computation being available. Inthe present work, the theory of discrete symmetries of the space of solutions tosDCHE is extended to the general case, apart from some natural exceptions. Definition and basic properties
Let us consider the following system of linear homogeneous first order ODEs z p “ ` µ ` p ℓ ´ q z ˘ p ´ q ` z r, (1) q “ ` λ ´ p ℓ ` q µz ˘ p ` µ q ` s, (2) z r “ ´ ` λ ` µ ˘ p ` z ` p ℓ ´ q ´ µz ˘ r ´ s, (3) z s “ ´ ` λ ` µ ˘ q ` z ` λ ´ p ℓ ` q µz ˘ r ` ` p ℓ ´ q z ´ µ ˘ s. (4)Here the symbols ℓ , λ , µ denote some complex constants. The symbols p, q, r, s stand for holomorphic functions of the complex variable z . For brevity, we shallrefer to them as pqrs -functions .When resolved with respect to the derivatives indicated by the symbol “ ” ,i.e. upon division of Eq.s (1)-(4) by z , , z , z , respectively, the coefficients intheir right-hand sides become rational functions holomorphic everywhere except,perhaps, at zero. Hence all solutions of the above system are holomorphic in thevicinity of any point z “ . Moreover, one may regard pqrs -functions as solu-tions of the Cauchy problem for Eq.s (1)-(4) with arbitrary (but not totally null)initial data specified at any given z “ z “ . Obviously, such local solution canbe analytically continued to any other point of C except zero. In particular, all solutions to Eq.s (1)-(4) (i.e. pqrs -functions) are single-valued holomorphic func-tions on any connected and simply connected subset of C ˚ “ C K t u containingany given z “ .At the same time it has to be noted that, except for the very special conditions,the natural (inextendible) domain of holomorphicity of pqrs -functions is neither C ˚ nor any its subset but rather the universal cover of C ˚ which is represented bycertain Riemann surface diffeomorphic to C , the covering projection Π : C ÞÑ C ˚ being realized by the natural exponential function. However, in what follows,we shall consider, unless otherwise specified, only a subdomain C ˚ of the noteddomain. It is convenient to represent it by the result of the removing from C ˚ of the ray R ´ of negative reals, C ˚ “ C ˚ K R ´ . Equivalently, C ˚ is the setconsisting of all complex numbers z such that either Im z “ or Im z “ & Re z ą . On C ˚ , any instance of pqrs -functions combines four single-valuedholomorphic functions uniquely defined by their values (which may be arbitrarybut all zero) at any given point z P C ˚ . The two their single-side continuationsto R ´ also exist giving rise to real analytic functions in the common domain but,as a rule, they do not coincide pointwise. Pqrs -functions reveal a number of noteworthy properties. The first of them isexpressed by the following statement.
Theorem 1.
Let pqrs -functions obey at z “ i the constraint q p i q ´ µp p i q ` r p i q “ . (5) Then the following equalities p p´ { z q “ ´ e i ℓπ z p ´ ℓ q p p z q , (6) q p´ { z q “ e i ℓπ z ´ ℓ ` µ p p z q ` z r p z q ˘ , (7) r p´ { z q “ e i ℓπ z p ´ ℓ q ` µz p p z q ` q p z q ˘ , (8) s p´ { z q “ ´ e i ℓπ z ´ ℓ ` µ ` µz p p z q ` q p z q ˘ ` z ` µz r p z q ` s p z q ˘˘ ; (9) hold true. Conversely, Eq. (5) follows from Eq.s (6) - (9) evaluated at z “ i . Remark 1.
Eq. (5) is obviously implied by Eq. (7) alone. The remaining threeequations, when evaluated at z “ i , either turn out to be fulfilled identically orfollow from Eq. (5) (and, thus, from Eq. (7)). Remark 2.
The constraint (5) does not affect the value of the function s atthe selected point and, moreover, s p¨q is present only in Eq. (9) which might beconsidered as decoupled from the preceding ones. However, there is an indirectinfluence of the selection of s (via the unrestricted setting of s p i q ) to the other pqr -functions in view of their “unbreakable interrelations” implied by Eq.s (1)-(4). Remark 3.
The involutive transformation of functions which we shall here referto as the transformation A , manifested in the the left-hand sides of Eq.s (6)-(9)
PECIAL FUNCTIONS ASSOCIATED WITH S DCHE 3 by the replacement z Ø ´ { z (10)of the argument z of the functions involved, is here tacitly regarded as the mapkeeping the particular argument z “ i unchanged. This point is worth mentioningbecause in the case we deal with, i.e. for functions possessing domains distinct of C ˚ , “the reflected imaginary unit” ´ i is not a fixed point of the implied transfor-mation of argument albeit ´ {p´ i q “ p´ i q , formally. Moreover, there is another transformation ˜A (or, one might say, another implementation of the rule (10)),not utilized here, recognizing just ´ i , not ` i , as a fixed point of the domain of a(this time) ˜A -transformed functions.Accordingly, as long as we consider ` i as the fixed point of the transformation ofargument replacement (10), there exist connected and simply connected open setscontaining ` i and contained in the subdomain of pqrs -functions such that theirimages through the transformation A also contain ` i and are contained in thementioned subdomain. On them, the asserted relations expressed by Eq.s (6)-(9)are well defined. It is here preferable to consider the subdomain C ˚ (or, perhaps,some its subsets) only. The extension of Eq.s (6)-(9) to the whole domain of pqrs -functions (the universal cover of C ˚ ) by means of analytic continuation isobviously feasible.To clarify some specialties of the above interpretation, we consider the followingexample. Let z be continuously moving from i P C ˚ towards some x P R ` Ă C ˚ along a concave curve. Then ´ { z , also starting from ` i but further differingfrom z , is moving around zero in the opposite angular direction, arriving ulti-mately at ´ x ´ P R ´ which does not belong to C ˚ . Thus, when dragging further z across R ` inward the half-plane Im z ă , the corresponding ´ { z leaves C ˚ across ‘the upper edge’ of the cut along the ray R ´ . Notice that we may notconsider it entering C ˚ again through the lower cut edge disconnected from theupper one. This means that in the course of the above process the literal appli-cability of the formulas (6)-(9) breaks down on the ray of positive reals. Thus, toensure their meaningfulness, one is compelled to obey the restriction Im z ą .At the same time, it is obvious that such a limitation is only a consequence ofcertain simplification we had adopted for convenience. It would not arise in caseof consideration of pqrs -functions on their full domain. However, then yet anothercomplication related to certain non-uniqueness of interpretation of Eq.s (6)-(9)would appear. In total, we still prefer here to restrict consideration to the sub-domain C ˚ keeping in mind limitations introduced by such a simplification.The above speculation shows how, given z , the result of the transformationof arguments of the functions in the left-hand sides of Eq.s (6)-(9) has to beinterpreted. Namely, one has to get any “shortest” (with respect to variation ofthe central angle) curve avoiding zero and linking ` i and z . Since the argument z “ i is sent by the transformation A to itself, the A-transformed curve also beginsat ` i . It is then its end point which is adopted as the value of the transformed S.I. TERTYCHNIY argument of the functions on the left in Eq.s (6)-(9) displayed there as ´ { z .This algorithm yields a holomorphic transformation concordant with the rule(10) which sends functions holomorphic in connected and simply connected opensets containing ` i and not containing to functions holomorphic in similar opensets.Eq. (5) singles out some subset of pqrs -functions constraining their values (i.e.the initial data for Eq.s (1)-(4)) at z “ i . Yet another property of pqrs -functionsleans on their parameterizing by the values at z “ . It reads as follows. Theorem 2.
Let λ ` µ “ and pqrs -functions obey the constraints µ p p q ` q p q ` r p q “ , (11) λ p p q ´ µ q p q ` s p q “ . (12) Then the following equalities p p { z q “ ´p λ ` µ q ´ z p ´ ℓ q ` µz r p z q ` s p z q ˘ , (13) q p { z q “ ´p λ ` µ q ´ z ´ ℓ ` λz r p z q ´ µ s p z q ˘ , (14) r p { z q “ ´ z p ´ ℓ q ` µz p p z q ` q p z q ˘ , (15) s p { z q “ ´ z ´ ℓ ` λz p p z q ´ µ q p z q ˘ ; (16) hold true. Conversely, Eq.s (11) , (12) follow from Eq.s (13) - (16) evaluated at z “ . Remark 4.
The replacement of argument of the functions on the left in Eq.s (13)-(16) (we shall refer to it as the transformation C ) does not affect the point z “ .Accordingly, there exist the open sets containing ` which remain invariant underthe action of the transformation C . Then it is reasonable to consider first theequalities (13)-(16) on such neighborhoods of the unity and then utilize analyticcontinuation for their extending to greater domains. Remark 5.
Besides z “ , the point z “ ´ (excluded, by definition, from C ˚ ) is also unaffected by the replacement z Ø { z utilized in Eq.s (13)-(16),formally. However, it can not be considered as a fixed point of the transformationC. More precisely, claiming of z “ ´ to be a fixed point, one must replacethe above transformation C by “yet another implementation” ˜C of the argumenttransformation. For it, the former fixed point z “ loses such a property. Besides,for ˜C , the associated (sub-)domain of pqrs -functions, playing role of C ˚ , has tocontain R ´ but not R ` . Having thus noted the presence of certain ambiguity ininterpretation of Eq.s (13)-(16) we limit ourselves with the above remarks andshall not consider here this issue in greater details.The combination of conditions of the two above theorems yields one morerelationship in accordance with the following. PECIAL FUNCTIONS ASSOCIATED WITH S DCHE 5
Theorem 3.
Let pqrs -function obey the conditions of both Theorem 1 and The-orem 2, i.e. they are holomorphic on a connected and simply connected open setcontaining ` i and ` and meet the constraints (5) , (11) , and (12) . Then theequalities M { r p s “ e i ℓπ p λ ` µ q ´ ` µz r ` s ˘ , (17) M { r q s “ ´ e i ℓπ ` p µ z p ` q q ` µ p λ ` µ q ´ z p µz r ` s q ˘ , (18) M { r r s “ ´ e i ℓπ r, (19) M { r s s “ e i ℓπ ` p λ ` µ q p ` µz r ˘ , (20) hold true, where the arguments z of all the functions coincide and hence aresuppressed, and where the operator M { carries out analytic continuation of thefunction it acts to along the circular arc started at z , centered at zero, subtendingan angle π , and oriented counter-clockwise. Remark 6.
As opposed to transformations of pqrs -functions treated by Theorems1 and 2, the transformation of arguments of functions on the left in Eq.s (17)-(20)(let us call it the transformation B ) admits no fixed points and is not involutive.Moreover, applying the transformation B twice, the resulting effect turns into theanalytic continuation of the function to be transformed along the loop projectedto (essentially, coinciding with) the full circle . Such kind of analytic continuationaround a singular point (in our case, the center z “ ) is commonly named the monodromy transformation. We denote it by the symbol M . We have therefore M { ˝ M { “ M by definition. The effect of the operator M { can thus benamed semi-monodromy transformation.In our case M is the linear operator which sends, in particular, the values of pqrs -functions on the “lower” edge of the cut along the ray R ´ to the values theyassume on its “upper” edge. Since pqrs -functions obey on the both edges the samesystem (1)-(4) of linear homogeneous ODEs (their coefficients are invariant withrespect to M ) such a transformation is represented by a constant ˆ matrix. Remark 7. If z P C ˚ and Im z ă then M { z ` “ M { r Id sp z q ˘ “ e i π z “´ z P C ˚ . However, if Im z ě then an application of M { would yield theargument of pqrs -functions in Eq.s (17)-(20) on the left which does not belong to C ˚ . Evading such a complication, we shall assume Im z ă for simplicity unlessotherwise specified. Analytic continuation has to be applied for relaxation of thelimitation and extending them to a greater domain. Remark 8.
In general case, given a prescribed set of constant parameters, si-multaneous fulfillment of Eq. (5) and Eq.s (11), (12) for the same instance of pqrs -functions should be achievable by means of their appropriate selection. In-deed, the set of all pqrs -functions can be indexed by the quad of their values at z “ fixed up to multiplication by an insignificant (associated with a decoupleddegree of freedom) non-zero common factor, i.e. by points of a projective space S.I. TERTYCHNIY C P t u . The two linear equations (11),(12) single out the projective line embeddedtherein. This projective line is conveyed (pushedforward) by the vector flow as-sociated with the equations (1)-(4) into another projective space C P t i u indexingthe same set of pqrs -functions by their values (also considered up to a commonconstant factor) at z “ i . In the latter projective space, the equation (5) singlesout certain embedded projective plane. The question equivalent to the issue ofconsistency of Eq. (5) with Eq.s (11) and (12) reads: whether the former (con-veyed) projective line intersects the latter projective plane or not? This problemremains open yet but numerical computations point in favor of the affirmativeupshot, at least, under apparently generic conditions. Thus, most plausibly, in-consistency of Eq. (5) with Eq.s (11) and (12) and the subsequent emptiness ofTheorem 3, if any, could only occur under the very special conditions (currentlyunknown). Corollary 4.
There exists a set of pairwise linearly independent quads of holo-morphic functions p, q, r, s parameterized by points of C P such that the equations (6) - (9) are fulfilled. Corollary 5.
There exists a set of pairwise linearly independent quads of of holo-morphic functions p, q, r, s parameterized by points of C P such that the equations (13) - (16) are fulfilled. Conjecture .
For almost all values of the constant parameters there exists adiscrete set of quads of functions p, q, r, s holomorphic on the universal cover of C ˚ such that the equations (6) - (9) , (13) - (16) , and (17) - (20) are fulfilled. Now proceed now with proofs of the three above theorems.
Proof of Theorem 1.
Let us denote the four differences of the left- and right-handsides of Eq.s (6), (7), (8), (9), by the symbols A ∆ p , A ∆ q , A ∆ r , A ∆ s respectively,considering them, as they stand, as the functions of z . For example, one ofsuch definitions reads A ∆ p p z q “ p p´ { z q ` e i ℓπ z p ´ ℓ q p p z q , etc . As it is shown inAppendix A, they obey the following system of linear homogeneous ODEs z ddz A ∆ p “ z p ´ ℓ ` µz q A ∆ p ´ z A ∆ q ` A ∆ r ,z ddz A ∆ q “ pp ℓ ´ q µ ` λz q A ∆ p ` µz A ∆ q ` z A ∆ s ,z ddz A ∆ r “ ´ p λ ` µ q z A ∆ p ´ ` µ ` p ℓ ´ q z ˘ A ∆ r ´ z A ∆ s z,z ddz A ∆ s “ ´ p λ ` µ q z A ∆ q ` pp ℓ ` q µ ` λz q A ∆ r ´ z p ℓ ´ ` µz q A ∆ s , (21)provided Eq.s (1)-(4) are fulfilled.Using explicit definitions, let us compute the particular values of A ∆ ✪ p i q “ A ∆ ✪ p e i2 π q for ✪ “ p, q, r, s . Notice that for such a choice of the argument z one PECIAL FUNCTIONS ASSOCIATED WITH S DCHE 7 has ´ { z “ ´ e ´ i2 π “ i , z ´ ℓ “ e ´ i ℓπ , z p ´ ℓ q “ ´ e ´ i ℓπ . Then it follows fromEq. (6) that A ∆ p p i q “ . Other A ∆ ✪ p i q are not automatically zero but one easilyfinds that A ∆ ✪ p i q “ ζ ✪ ¨ ` q p i q ´ µp p i q ` r p i q ˘ for ✪ “ q, r, s, and ζ q “ ζ r “ , ζ s “ µ. Thus if Eq. (5) is fulfilled then A ∆ ✪ p i q “ for all ‘the indices’ ✪ “ p, q, r, s .This implies the vanishing everywhere of all A ∆ ✪ p z q in view of the uniqueness ofsolutions of the Cauchy problem for Eq.s (21) with the null initial data posed at z “ i . (cid:3) Proof of Theorem 2.
Building on the notations utilized in the preceding proof,we denote the differences of the left- and right-hand sides of Eq.s (13), (14), (15),(16) by the symbols C ∆ p p z q , C ∆ q p z q , C ∆ r p z q , C ∆ s p z q , respectively. It is shown inAppendix B that they obey the following system of linear homogeneous ODEs z ddz C ∆ p “ ´ z p ℓ ´ ` µz q C ∆ p ` z C ∆ q ´ C ∆ r ,z ddz C ∆ q “ pp ℓ ` q µ ´ λz q C ∆ p ´ µz C ∆ q ´ z C ∆ s ,z ddz C ∆ r “ p λ ` µ q z C ∆ p ` p µ ` p ´ ℓ q z q C ∆ r ` z C ∆ s ,z ddz C ∆ s “ p λ ` µ q z C ∆ q ` pp ℓ ` q µ ´ λz q C ∆ r ` z p ´ ℓ ` µz q C ∆ s , (22)provided Eq.s (1)-(4) are fulfilled.We compute now the values of the differences C ∆ ✪ for ✪ = p, q, r, s , after plugging z Ø in their definitions. The result is as follows: p λ ` µ q C ∆ ✪ p q “ ζ ✪ ¨ ` s p q ´ µq p q ` λp p q ˘ ` σ ✪ ¨ ` r p q ` q p q ` µp p q ˘ , where ζ p “ , ζ q “ ´ µ, ζ r “ , ζ s “ λ ` µ ,σ p “ µ, σ q “ λ, σ r “ λ ` µ , σ s “ . Thus if the constraints (11) and (12) are fulfilled then all the differences C ∆ ✪ p z q vanish at z “ . But then they are the identically zero functions, C ∆ ✪ p z q ” , asa consequence of Eq.s (22). This means exactly that Eq.s (13)-(16) hold true. (cid:3) Proof of Theorem 3.
As above, let us denote the differences of the left- and right-hand sides of Eq.s (17), (18), (19), (20) by the symbols B ∆ p p z q , B ∆ q p z q , B ∆ r p z q , B ∆ s p z q ,respectively. It is shown in Appendix C that in case of fulfillment of Eq.s (1)-(4)they obey the following system of linear homogeneous ODEs z ddz B ∆ p “ pp ℓ ´ q z ´ µ q B ∆ p ` B ∆ q ´ z B ∆ r ,ddz B ∆ q “ ´ p λ ` p ℓ ` q µz q B ∆ p ´ µ B ∆ q ´ B ∆ s , (23) S.I. TERTYCHNIY z ddz B ∆ r “ p λ ` µ q B ∆ p ` p p ℓ ´ q ` µz q z B ∆ r ` B ∆ s ,z ddz B ∆ s “ p λ ` µ q B ∆ q ´ p λ ` p ℓ ` q µz q z B ∆ r ` p µ ` p ℓ ´ q z q B ∆ s . The next step should assume computation of the particular values B ∆ ✪ p´ i q , ✪ “ p, q, r, s . However, carrying out this by means of the mere substitutions z Ø ´ i into the definitions of B ∆ ✪ , some ambiguity may arise due to possibility of over-lapping of sheets of the branching domain pqrs -functions live on. To make thecomputation univocal, we consider first the “deformed” versions ǫB ∆ ✪ of the dif-ferences B ∆ ✪ . Their distinction is that in case of ǫB ∆ ✪ the factor in argument ofthe pqrs -function on the left is distinct of the one involved in Eq.s (17) - (20),see Remark 7. The common exponential multiplier on the right is also modified.Namely, let the factor e i ǫπ , where ǫ P r , s is an auxiliary real parameter, beused instead of ´ “ e i π . For example, one has ǫB ∆ r p z q “ r p e i ǫπ z q ` e i ℓǫπ r p z q while B ∆ r p z q “ r p e i π z q ` e i ℓπ r p z q , etc . Explicit definitions of all ǫB ∆ ✪ are given byEq.s (58).Now let us notice that in the case ǫ “ all the arguments of pqrs -functionsutilized for computation of B ∆ ✪ p z q coincide with z and no ambiguity can thusarise. Then, starting from these values, we carry out analytic continuation, vary-ing ǫ through the segment r , s , and define B ∆ ✪ p z q to be “the final values” thefunctions ǫB ∆ ✪ p z q arrive at as ǫ Õ . Such an interpretation leaves no room forambiguity in the meaning of definition of B ∆ ✪ and, more generally, the relationsEq.s (17)-(20) represent.Assuming the above interpretation of B ∆ ✪ , it is shown in Appendix D that thefollowing equations are fulfilled for arbitrary functions p, q, r, s holomorphic onthe circular arc passing through the point ´ i , ` , and ` i . B ∆ p p´ i q “ e i ℓπ p λ ` µ q ´ ` µ C ∆ r p i q ´ C ∆ s p i q ˘ , B ∆ q p´ i q “ ` q p i q ´ µ p p i q ` r p i q ˘ ` e i ℓπ ` C ∆ q p i q ´ µ C ∆ p p i q ` µ p λ ` µ q ´ p µ C ∆ r p i q ´ C ∆ s p i qq ˘ , B ∆ r p´ i q “ ` q p i q ´ µ p p i q ` r p i q ˘ ` e i ℓπ C ∆ r p i q , B ∆ s p´ i q “ µ ` q p i q ´ µ p p i q ` r p i q ˘ ´ e i ℓπ p λ ` µ q C ∆ p p i q ` µ e i ℓπC ∆ r p i q . (24)The symbols C ∆ ✪ were already utilized in the proof of Theorem 2. They denotethe differences of the left- and right-hand sides of Eq.s (13)-(16), considered, asthey stand, as the functions of z . Each of Eq.s (24) can therefore be regarded asthe coincidence, upon simplifications, of a pair of certain linear combinations of4+4 instances of pqrs -functions evaluated at z “ i and at z “ ´ i .On the other hand, the conditions of the theorem to be proven imply, in partic-ular, the fulfilment of the assertion of Theorem 2 which establishes the vanishing PECIAL FUNCTIONS ASSOCIATED WITH S DCHE 9 of all the four functions C ∆ ✪ p z q irrespectively of the value of their argument. Thusall the terms in Eq.s (24) involving those factors may be discarded.Now, taking into account the fulfillment of Eq. (5), we see that all the ex-pressions on the left in (24), i.e. the functions B ∆ ✪ p z q , ✪ = p, q, r, s , evaluated at z “ ´ i q , actually vanish. Since these functions obey the system of linear homoge-neous first order ODEs (see Eq.s (23)) they reduce to identical zero. This meansexactly that the equalities (17)-(20) hold true. (cid:3) The first integral
It proves sometimes to be useful to take into account the following noteworthyproperty of solutions to Eq.s (1)-(4).
Theorem 6.
Let λ ` µ “ . Then the following statements hold true. (0) If holomorphic functions p, q, r, s obey Eq.s (1) - (4) then the value of ex-pression D “ z p ´ ℓ q ` p p z q s p z q ´ q p z q r p z q ˘ (25) does not depends on z ; (1) if holomorphic functions p, q, r, s obey Eq.s (6) - (9) then Y z Ø ´ { z D u “ t D u , (26) where and in what follows t D u denotes the right-hand side of Eq. (25) considered as a function of z ; (2) if holomorphic functions p, q, r, s obey Eq.s (13) - (16) then Y z Ø { z D u “ t D u ; (27)(3) if holomorphic functions p, q, r, s obey Eq.s (17) - (20) then Y z Ø M { z D u “ t D u . (28)It has to be added that the precise meaning of the argument replacements z Ø ´ { z , z Ø { z , and z Ø M { z p» ´ z q involved in the above formulas is thesame as in the corresponding systems of the equations claimed to be fulfilled. Proof.
We shall consider the above assertions one by one.Assertion (0). Let us expand the expression of the derivative of the right-hand sideof Eq. (25) in case of arbitrary holomorphic functions p, q, r, s . A straightforwardcomputation establishes validity of the following identity z ℓ dd z t D u ” s ∆ p ´ z r ∆ q ´ q ∆ r ` p ∆ s . (29)Here the symbols ∆ ✪ p z q , where ✪ “ p, q, r, s, denote the differences of the left-and right-hand sides of Eq.s (1), (2), (3), (4), respectively, as they stand. Hence if the latter equations are fulfilled then the derivative (29) vanishes and t D u doesnot depend on z .Assertion (1). Its validity follows from the equality Y z Ø ´ { z D u “ t D u ` e ´ i ℓπ v e ´ i ℓπ z p ℓ ´ q ` s p´ { z q A ∆ p p z q ´ r p´ { z q A ∆ q p z q ˘ ´ ` r p z q ` µ z ´ p p z q ˘ A ∆ r p z q ´ p p z q A ∆ s p z q w , (30)holding true for arbitrary functions p, q, r, s , holomorphic at (and in the vicinityof) z “ i . Here the symbols A ∆ ✪ , ✪ “ p, q, r, s , denote the differences of the left-and right-hand sides of Eq.s (6), (7), (8), (9), respectively, as they stand.Eq. (30) follows, in turn, from the identity (65) given in Appendix E.Thus if Eq.s (6)-(9) are fulfilled then the equality (26) holds true.Assertion (2). Let us consider the following identity Y z Ø { z D u ” t D u ` z p ℓ ´ q ` s p { z q C ∆ p p z q ´ r p { z q C ∆ q p z q ˘ ` p λ ` µ q ´ ` p λr p z q ´ µ z ´ s p z qq C ∆ r p z q´ p µ z r p z q ` s p z qq C ∆ s p z q ˘ (31)which is verifiable by straightforward computation. Here C ∆ ✪ p z q , ✪ “ p, q, r, s denote the differences of the left- and right-hand sides of Eq.s (13),(14),(15),(16),respectively, considered, as they stand, as the functions of z . The equality (31)holds true for arbitrary functions p, q, r, s holomorphic at (and in the vicinity of) z “ . It is extended to any other z by means of analytic continuation.In view of (31), it is obvious that if Eq.s (13)–(16) are fulfilled then Eq. (27)holds true.Turning to the assertion (3), let us consider the equation Y z Ø M { z D u “ t D u ` e ´ i ℓπ z p ´ ℓ q v e ´ i ℓπ p ð s B ∆ p ´ ð r B ∆ q q ` p µ z p ` q q B ∆ r ` p λ ` µ q ´ p µ z r ` s qp µ z B ∆ r ` B ∆ s q w . (32)Here the symbols B ∆ ✪ , where ✪ “ p, q, r, s , denote the differences of the left-and right-hand sizes of the equations (17), (18), (19), (20), respectively. ‘Thediacritic mark’ ð denotes the transformation of the function argument definedas follows: ð ✪ “ ð ✪ p z q “ lim ǫ Õ ✪ p e i ǫπ z q . Here lim ǫ Õ should be understood as theanalytic continuation along the image of the segment r , s Q ǫ to the end pointcorresponding to ǫ “ . In Theorem 3 such a transformation is associated withthe operator M { .Eq. (32) follows from the identity (66) given in Appendix E. In turn, underconditions of the theorem, Eq. (28) is the obvious consequence of Eq. (32). (cid:3) The constant D (the first integral for the system (1)-(4) in fact) may vanish.Indeed, if it is null at some point (this is a quadratic constraint to values of pqrs -functions thereat) then it is zero everywhere. Such a case bears many signsof a degeneracy — being nevertheless in no way meaningless. Following here the PECIAL FUNCTIONS ASSOCIATED WITH S DCHE 11 requirement of genericity, we assume throughout that D “ without separatementioning. It is also worth noting that there is another case for which many ofthe relations discussed here degenerate. Namely, this takes place if λ ` µ “ .We evade here clarification of its specialties as well.3. On applications of pqrs -functions
The properties of pqrs -functions established above make them an object ofnotable interest by itself. However, they arose originally in the context of studyof symmetries of the space of solutions to the following ordinary second-orderlinear homogeneous differential equation z E ` ` p ℓ ` q z ` µ p ´ z q ˘ E ` ` ´ p ℓ ` q µ z ` λ ˘ E “ . (33)Here E “ E p z q is the unknown holomorphic function, ℓ, λ, µ are the constantparameters which can be identified with those involved in Eq.s (1)-(4).Eq. (33) belongs to the family of double confluent Heun equations ( DCHE ).These equations are discussed in Ref.s [1, 2], Ref.s [3], [4] contain some morerecent bibliography. Since a generic
DCHE is characterized by the four constantparameters, whereas Eq. (33) involves only three ones, Eq. (33) was named a special double confluent Heun equation (s
DCHE ). This reference is utilized in thepresent work as well.It should be noted that Eq. (33) was segregated within the
DCHE family be-cause of its intimate relation (in fact, equivalence) to the following nonlinearfirst-order ODE ϕ ` sin ϕ “ B ` A cos ωt, in which ϕ “ ϕ p t q is the unknown function, the symbols A, B, ω denote realconstants, t is a free real variable, and the dot denotes the derivative with respectto t . The latter equation and its generalizations are, in turn, well known due totheir emerging in a number of problems in physics (most notably in the modelingof a Josephson junction) [5, 6, 7], mechanics [8, 9], dynamical systems theory [10],and geometry [11].In earlier investigations the functions, obeying equations equivalent to Eq.s (1)-(4), were utilized for the constructing of linear transformations mapping the spaceof solutions to Eq. (33) into itself [12]. It was found that such transformationsdetermine a group which can be regarded as a discrete symmetry of the notedspace of solutions. (More precisely, in case of real parameters, one of three groupsarises depending on their values).The principal limitation of those considerations was, however, the restrictionof the parameter ℓ from Eq. (33) (sometimes called the order of this equation)to integers only. The simplification following from this assumption (the startingpoint of derivation of the mentioned symmetry transformations, in fact) is the reducing of the functions equivalent to our pqrs -functions to polynomials in z as well as in the parameters λ, µ . Moreover, there exists the recurrent schemeenabling one to compute these polynomials for any given positive integer ℓ .The definition of pqrs -functions considered in the present work needs no sucha restriction that enables us to make a crucial step in revealing of discrete sym-metries of the noted space of solutions in case on non-integer ℓ . We apply theapproach closely following the one utilized in the case of integer order althoughsome specific subtleties still have to be taken into account.To that end, let us define the two families of linear operators, ǫ L A and ǫ L B ,depending on the real parameter ǫ P r´ , s . They act to arbitrary functions(denoted E ) holomorphic in C ˚ in accordance with the following formulas. ǫ L A r E sp z q “ e µ p z ` { z q Y z Ø e i ǫπ { z z p p z q E p z q ` q p z q E p z q \ , (34) ǫ L B r E sp z q “ z ´ ℓ e µ p z ` { z q Y z Ø e i ǫπ z z r p z q E p z q ` s p z q E p z q \ . (35)The functions p, q, r, s are assumed to be holomorphic in the same domain.If ǫ “ then the common argument of the functions p, q, E and r, s, E in right-hand sides of (34) and (35) coincide with { z and z , respectively. Then the valuesof L A r E s and L B r E s are correctly defined everywhere in C ˚ .We introduce now the particular instances L A and L B of the operators ǫ L A and ǫ L B as the results of the analytic continuations, starting from L A and L B , alongthe images of the segment r , s Q ǫ through the corresponding maps z ÞÑ e i ǫ { z and z ÞÑ e i ǫ z . We may write down these relationships as follows. L A “ lim ǫ Õ ǫ L A , L B “ lim ǫ Õ ǫ L B . (36) Theorem 7.
Let pqrs -functions verify Eq.s (1) - (4) and E verify Eq. (33) . Thenthe functions L A r E s and L B r E s also verify Eq. (33) .Proof. Let us introduce the operator H associated with Eq. (33), i.e. let H r E sp z q “ z E p z q` ` p ℓ ` q z ` µ p ´ z q ˘ E p z q` ` λ ´ µ p ℓ ` q z ˘ E p z q . (37)Composing it with the operator L A , the following expansion of the slightly mod-ified result of their combined action to an arbitrary holomorphic function E can The positive integer order ℓ P Z ` determines their degrees linearly increasing with itsincrement. As to the case of negative ℓ P Z ´ , there is a trick enabling one to convert it to thecase of the positive order equal to | ℓ | . PECIAL FUNCTIONS ASSOCIATED WITH S DCHE 13 be obtained Z z Ø ´ { z e ´ µ p z ` { z q p H ˝ L A qr E s \ “ z p H r E s` ` µ p ´ q ` z r ˘ H r E s` z E ∆ p ` z E ∆ q ` ` p λ ´ p ℓ ` q µz q E ` z E ` p µ p ´ z q ` z q E ˘ ∆ p ` ` p µ p ´ z q ´ z p ℓ ´ ´ µz qq E ` z E ˘ ∆ q ` z E ∆ r ` E ∆ s , (38)provided the functions E, p, q, r, s of the variable z are holomorphic at (and inthe vicinity of) z “ i . In Eq. (38) the symbols ∆ ✪ p z q , where ✪ “ p, q, r, s, denotethe differences of the left- and right-hand sides of Eq.s (1)-(4) (they were alreadyused in the proof of Theorem 6), H “ d { dz ˝ H .In case of the operator L B similar expansion looks as follows. z ℓ ´ e ´ µ p z ` { z q p H ˝ L B qr E s “ z ð r ð H r E s` ` p ℓ ´ q z ð r ` ð s ` z ð r ˘ ð H r E s` z ð E ∆ r ` ð E ∆ s ´ p λ ` µ q ` ð E ∆ p ` ð E ∆ q ˘ ´ ` p µ ´ p ℓ ´ q z q ð E ` p λ ` p ℓ ` q µz q ð E ˘ ∆ r `p ð E ´ µ ð E q ∆ q . (39)The symbols ∆ ✪ have the same meaning as in Eq. (38). ‘The diacritic mark’ ð , denoting the semi-monodromy transformation, was also used in the proofof Theorem 6. It is worth reminding that ð ✪ “ ð ✪ p z q “ lim ǫ Õ ✪ p e i ǫπ z q for anyholomorphic function ✪ . If the functions E, p, q, r, s are holomorphic on C ˚ and Im z ă then the argument of evaluation of ‘semi-monodromy-transformed’functions also belong to C ˚ and all the constituents of Eq. (39) are well defined.Other values of z are to be handled by means of analytic continuation.The equalities (38), (39) follow from the identities (67) and (68), respectively,given in Appendix F. In turn, the theorem’s assertion follows from Eq.s (38)and (39) since the fulfillment of Eq.s (1)-(4) implies ∆ ✪ “ and the identicalvanishing of H r E s is equivalent to Eq. (33) which is also assumed to be fulfilled. (cid:3) Remark 9.
The transformations realized by the operators L A and L B carry outthe (lifted) replacements z Ø ´ { z and z Ø ´ z of arguments of the functionsinvolved. There exists the third operator which we denote L C also sending anysolution to Eq. (33) to some its solution and utilizing the missing replacement z Ø { z of arguments which in conjunction with the preceding ones constitute theKlein group of maps naturally acting on C ˚ . L C is not linked to pqrs -functionsand is well defined for any choice of constant parameters. It can be represented by the following formula. L C r E sp z q “ z ´ ℓ ´ Y z Ø { z E p z q ´ µE p z q u . (40)In view of the non-trivial structure of the domain of solutions to Eq. (33) “theimplementation” of the rule (40) is not unique. In particular, for one of them (thelift of) ` is the fixed point of the map indicated by the argument replacement z Ø { z whereas for the other one it is (the lift of) ´ which plays a similar role.The transformations of the space of solutions to Eq. (33) associated with pqrs -functions possess the properties of quasi-involutions similar to ones found earlierin the case of integer order ℓ , cf Ref. [13], Eq.s (34), (35).
Theorem 8.
Let λ ` µ “ , the function E obey Eq. (33) , and pqrs -functionsobey Eq.s (1) - (4) . If, additionally, (1) Eq.s (6) - (9) hold true then p L A ˝ L A qr E s “ ´ e i ℓπ D ¨ E ; (41)(2) Eq.s (17) - (20) hold true then p L B ˝ L B qr E s “ ´p λ ` µ q e ℓπ D ¨ M r E s . (42) Proof.
The above claims follow from the equalities p L A ˝ L A qr E sp z q ` e i ℓπ t D u E p z q “ ` p s p z q ` µ z ´ q p z qq E p z q` p z r p z q ` µ p p z qq E p z q ˘ A ∆ p p z q` ` q p z q E p z q ` z p p z q E p z q ˘ A ∆ q p z q` ð p p z ´ q ` p p z q H r E sp z q` E p z q ∆ p p z q ` E p z q ∆ q p z q ˘ , (43) e i ℓπ z p ℓ ´ q v p L B ˝ L B qr E s ` p λ ` µ q e ℓπ t ð D u ö E w “´p λ ` µ q ð r ¨ ` z ö E ð B ∆ p ` ö E ð B ∆ q ˘ ´p µ z ð r ` ð s q ` z ö E ð B ∆ r ` ö E ð B ∆ s ˘ ` ð r ¨ ` z ö r ö H ` z ö E ö ∆ r ` ö E ö ∆ s ˘ , (44)involving arbitrary functions E, p, q, r, s and their derivatives. These are, in turn,the consequences of the identities (69) and (70), given in Appendix G. Concerningthe notations utilized therein, let us remind that t D u denote the right-hand sideof Eq. (25) considered as a function of z . The symbols ∆ ✪ , A ∆ ✪ , and B ∆ ✪ ,where ✪ “ p, q, r, s , denote the differences of the left- and right-hand sides forEq.s (1)-(4), for Eq.s (6)-(9), and for Eq.s (17)-(20), respectively. They are alsoconsidered as the functions of z . PECIAL FUNCTIONS ASSOCIATED WITH S DCHE 15
There are also the two kinds of ‘diacritic marks’ in use. Of them, ‘the accent’ ð indicates the transformation of the function argument carrying out its continuousanti-clockwise rotation in the complex plane at an angle π . It was earlier namedthe semi-monodromy map. In Theorem 3 such a transformation is associated withthe operator M { . Evidently, if Im z ă then ð ✪ p z q is simply ✪ p´ z q . However,if Im z ě then the semi-monodromy transformation sends such argument outthe subdomain C ˚ and hence it can not be expressed by the inversion of thesign. It worth noting here that ð p p z ´ q (see the last but one line in Eq. (43)) iswell defined, provided Im z ą . Indeed, then Im z ´ ă and the argument ofevaluation of the function p when computing ð p p z ´ q “ lim ǫ Õ p p e i ǫπ z ´ q belongsto C ˚ .The second ‘accent’ ö has a similar meaning but “the rotation angle” of afunction argument is here twice as much amounting to π . Such a transformationlooks like a full revolution but it does not lead to the identical map in view ofnon-trivial structure (distinction of complex plane or any subset of the complexplane) of the domains of the functions we consider. Rather it corresponds to themonodromy transformation.For some reasons we had agreed above to consider pqrs -functions on their sub-domain C ˚ . Here, however, this is not enough and we are forced to introduce fora time a somehow extended one. Indeed, if z P C ˚ then the point of evaluation ofa monodromy-transformed function does not belong to C ˚ due to the cut alongthe ray of negative reals which the path of analytical continuation inevitablymeets. “The minimally extended subdomain” where the monodromy map canstill be consistently defined is constructed, for instance, by means of addition ofanother copy of C ˚ and the gluing of it to the original one along the oppositeedges of their cuts (the two complementary ones remain free). Then if z belongsto the “lower” (original) sheet of this ‘double - C ˚ ’ the point of evaluation of an-alytic continuation of the function to be monodromy transformed belongs to theupper one and in this way all the constituents of Eq. (44) can be computed (andit is finally fulfilled).The important circumstance is, however, that under conditions of the theoremthe evaluation of many functions and the handling of the associated subtletiesit implies is superfluous. Indeed, the fulfillment of certain equations required bythe theorem conditions means the vanishing of the expressions H r E s , ∆ ✪ , A ∆ ✪ , and B ∆ ✪ which yield zero independently of the point of their evaluation. Theyconstitute a full collection of the factors in the right-hand sides of Eq.s (43),(44) such that if all they are zero then all the terms on the right vanish. Theonly function with transformed argument which ‘survives’ is the monodromytransformed function E involved in the left-hand side of Eq. (44) in the form ö E p“ M r E sq . Besides, we know that the fulfillment of Eq.s (1)-(4) leads to theindependence of t D u on the point of evaluation, see Theorem 6. Thus we may replace it, as well as t ð D u , with the corresponding constant D (whose value isactually determined in a complicated way by the parameters ℓ, λ, µ ).Now, as the right-hand sides of Eq.s Eq.s (43), (44) are zero, the vanishing oftheir left-hand sides leads just to Eq.s (41) and (42). (cid:3) Corollary 9.
If the conditions of Theorem 8 are fulfilled then the operators L A , L B it concerns determine automorphisms of the space of solutions to Eq. (33) . Summary
We define a family of quads of holomorphic functions (referred to, for brevity,as pqrs -functions) as the non-trivial solutions to the system of linear homogeneousfirst order ODEs (1)-(4). Each instance of such functions can be constructed asa solution of the Cauchy problem for the initial data specified at any given point z except zero. It is shown that, fixing the initial data at z “ i and claimingfulfillment of the linear homogeneous constraint (5), one obtains pqrs -functionswhich obey the equalities (6)-(9) (Theorem 1). Similarly, if the initial data arespecified at z “ and obey thereat the two linear homogeneous constraints (11),(12) then pqrs -functions obey the equalities (13)-(16) (Theorem 2). Lastly, if allthe three mentioned linear constraints (imposed at two distinct locations) aremet then the equalities (17)-(20) involving semi-monodromy map take place aswell (Theorem 3). Pqrs -functions had found application (if fact, arose) in frameworks of investi-gation of properties of solutions to special double confluent Heun equation (33).Under conditions here assumed the operators L A , L B defined by the formulas(36), (34), (35) turn out to define the automorphisms on the space of its solutions(Theorem 7). Moreover, they possess quite remarkable composition properties.It particular, the operator L A is “almost involutive” (see Theorem 8, Eq. (41))while L B , being applied twice, reduces, up to a known constant factor, to themonodromy transformation (Eq. (42)). Besides, they define automorphisms.In the special case of integer values of the constant parameter ℓ the functionsalmost identical to our pqrs -functions were originally introduced in Ref. [12]. Thedistinction of functions with the same notations considered therein against thepresent ones reduces to different normalizations of the functions p and q . Itis worth mentioning that the variant of pqrs -functions considered in [12] dealsexclusively with polynomials. Moreover, they are polynomial not only in z butalso in the parameters λ and µ (while ℓ determines the polynomial degrees).Thus we may claim that in the case of a (positive) integer ℓ Eq.s (1)-(4) admit apolynomial solution.
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Appendix A. Identities leading to Eq.s (21)Eq.s (21) the proof of Theorem 1 leans on are the straightforward consequencesof the four identities displayed below which are, in principle, verifiable by explicitcomputations . Namely, the following equalities hold true z ddz ǫA ∆ p p z q ” ´ z p ℓ ´ ` e ´ i ǫπ µz q ǫA ∆ p p z q` e ´ i ǫπ z ǫA ∆ q p z q ´ e i ǫπ ǫA ∆ r p z q´ e ´ i ǫπ z ∆ p p e i ǫπ { z q ` e i ǫℓπ z p ´ ℓ q ∆ p p z q`p ` e ´ i ǫπ q e i ǫℓπ z p ´ ℓ q ˆ ` µ p ` p ´ e i ǫπ q z q p p z q ´ e i ǫπ q p z q ` z r p z q ˘ , (45) z ddz ǫA ∆ q p z q ” e i ǫπ p e i ǫπ p ℓ ` q µ ´ λz q ǫA ∆ p p z q´ e i ǫπ µz ǫA ∆ q p z q ´ e i ǫπ z ǫA ∆ s p z q´ e i ǫπ z ∆ q p e i ǫπ { z q ´ e i ǫℓπ z ´ ℓ ` µ ∆ p p z q ` z ∆ r p z q ˘ As a matter of fact, they were handled with help of the computer algebra. Similar remarksconcern the majority of formulas in the present paper or, at least, all more or less lengthy ones. `p ` e i ǫπ q e i ǫℓπ z ´ ℓ ˆ ` pp λ ` µ q z ` p ´ e i ǫπ qp ℓ ` q µz ´ µ q p p z q` µq p z q ` z p µ p z ´ q r p z q ` s p z qq ˘ , (46) z ddz ǫA ∆ r p z q ” e ´ i ǫπ p λ ` µ q z ǫA ∆ p p z q ` e ´ i ǫπ z ǫA ∆ s p z q` ` e i ǫπ µ ´ p ℓ ´ q z ˘ ǫA ∆ r p z q´ e ´ i ǫπ z ∆ r p e i ǫπ { z q ´ e i ǫℓπ z p ´ ℓ q ` µ ∆ p p z q ` ∆ q p z q ˘ ´p ` e ´ i ǫπ q e i ǫℓπ z p ´ ℓ q ˆ ` p λ ` p ´ e i ǫπ q µ q z p p z q ` p ´ e i ǫπ q µq p z q` z p µz r p z q ` s p z qq ˘ , (47) z ddz ǫA ∆ s p z q ” e ´ i ǫπ p λ ` µ q z ǫA ∆ q p z q´ e i ǫπ p λz ´ e i ǫπ p ℓ ` q µ q ǫA ∆ r p z q´ z p ℓ ´ ´ e ´ i ǫπ µz q ǫA ∆ s p z q´ e ´ i ǫπ z ∆ s p e i ǫπ { z q` e i ǫℓπ z ´ ℓ ` µ ∆ p p z q ` µ ∆ q p z q ` µz ∆ r p z q ` ∆ s p z q ˘ `p ` e ´ i ǫπ q e i ǫℓπ z p ´ ℓ q ˆ ` µz ` λ ` µ ´ p e i ǫπ λ ` µ q z ` µe i ǫπ ` e i ǫπ ´ qp ℓ ` q z ˘ p p z q` p e ǫπ p ℓ ` q µ ´ µ z ´ e i ǫπ pp ℓ ` q µ ` λz q ˘ q p z q` z ` p λ ` µ p ´ z qq r p z q ´ µs p z q ˘˘ . (48)The symbols ∆ ✪ p z q , where ✪ “ p, q, r, s, stand for the differences of the left- andright-hand sides of the equations (1), (2), (3), (4), respectively. The symbol ǫ denotes the real parameter, ǫ P r´ , s . The definitions of the functions ǫA ∆ ✪ p z q , where ✪ “ p, q, r, s, read ǫA ∆ p p z q “ p p e i ǫπ { z q ` e i ǫℓπ z p ´ ℓ q p p z q , ǫA ∆ q p z q “ q p e i ǫπ { z q ´ e i ǫℓπ z ´ ℓ ` µ p p z q ` z r p z q ˘ , ǫA ∆ r p z q “ r p e i ǫπ { z q ´ e i ǫℓπ z p ´ ℓ q ` µz p p z q ` q p z q ˘ , ǫA ∆ s p z q “ s p e i ǫπ { z q ` e i ǫℓπ z ´ ℓ ` µ ` µz p p z q ` q p z q ˘ ` z ` µz r p z q ` s p z q ˘˘ . (49)Hence, as a matter of fact, the equalities (45)-(48) signify the four pairwise coin-cidences, upon simplification, of certain expressions constructed in two different PECIAL FUNCTIONS ASSOCIATED WITH S DCHE 19 ways from arbitrary holomorphic functions p, q, r, s and their first order deriva-tives.Let us notice now that in the limit ǫ Õ the functions ǫA ∆ ✪ p z q coincide withthe functions A ∆ ✪ p z q introduced in the beginning of the proof of Theorem 1and involved in Eqs (21). It is worth reminding that they were defined as thedifferences of the left- and right-hand sides of the equations (6)-(9). They arecorrectly defined if the functions p, q, r, s are holomorphic in the vicinity of z “ i .Besides, for ǫ “ , the last summands in the right-hand sides of the equalities (45)-(48), which are proportional to either p ` e i ǫπ q or p ` e ´ i ǫπ q , vanish. Finally, itremains to note that if the functions p, q, r, s obey Eq.s (1)-(4) then the differences ∆ ✪ p z q become identically zero for all ✪ “ p, q, r, s and all the summands whichcontain them can also be dropped out. After such simplifications, comparing theresulting form of Eq.s (45)-(45) with Eq.s (21), one easily finds that they coincide.Thus the equalities (21) hold true. Appendix B. Identities leading to Eq.s (22)Eq.s (22) follow from the identities given below which can be, in principle, ver-ified by straightforward computations. Namely, for any functions p, q, r, s holo-morphic, at least, in the vicinity of z “ the following equalities hold true z ddz C ∆ p p z q ” ´ z p ℓ ´ ` µz q C ∆ p p z q ` z C ∆ q p z q ´ C ∆ r p z q` z ∆ p p { z q ´ p λ ` µ q ´ z p ´ ℓ q p µz ∆ r p z q ` ∆ s p z qq , (50) z ddz C ∆ q p z q ” pp ℓ ` q µ ´ λz q C ∆ p p z q ´ µz C ∆ q p z q ´ z C ∆ s p z q` z ∆ q p { z q ´ p λ ` µ q ´ z ´ ℓ p λz ∆ r p z q ´ µ ∆ s p z qq , (51) z ddz C ∆ r p z q ” p λ ` µ q z C ∆ p p z q ` p µ ` p ´ ℓ q z q C ∆ r p z q ` z C ∆ s p z q` z ∆ r p { z q ´ z p ´ ℓ q p µ ∆ p p z q ` ∆ q p z qq , (52) z ddz C ∆ s p z q ” p λ ` µ q z C ∆ q p z q ` pp ℓ ` q µ ´ λz q C ∆ r p z q` z p ´ ℓ ` µz q C ∆ s p z q` z ∆ s p { z q ´ z ´ ℓ p λ ∆ p p z q ´ µ ∆ q p z qq . (53)Here the symbols C ∆ ✪ p z q , where ✪ “ p, q, r, s, used already in the proof of Theo-rem 2, stand for the differences of the left- and right-hand sides of Eq.s (13)-(16).The symbols ∆ ✪ p z q , where ✪ “ p, q, r, s, denote the differences of the left- andright-hand sides of the equations (1)-(4). Thus the equalities (50)-(53) signifythe pairwise coincidences, upon simplification, of certain expressions constructedin two different ways from arbitrary holomorphic functions p, q, r, s and their firstorder derivatives. Obviously, these expressions are correctly defined if the abovefour functions p, q, r, s are holomorphic in the vicinity of z “ . Finally, if the functions p, q, r, s are not arbitrary but verify Eq.s (1)-(4) thenthe differences ∆ ✪ p z q vanish and the identities (50)-(53) convert to Eq.s (22)which are therefore the direct consequence of Eq.s (1)-(4). Appendix C. Identities leading to Eq.s (23)Eq.s (23), utilized in the proof of Theorem 3, can be obtained from the fouridentities displayed below which are verifiable by straightforward computations.Namely, it can be shown that z ddz ǫB ∆ p p z q ” p´ µ ` p ℓ ´ q z q ǫB ∆ p p z q ´ e ´ i ǫπ ǫB ∆ q p z q ´ z ǫB ∆ r p z q` e ´ i ǫπ ∆ p p e i ǫπ z q ´ e i ℓǫπ p λ ` µ q ´ p µz ∆ r p z q ` ∆ s p z qq`p ` e ´ i ǫπ q ` µ p p e i ǫπ z q ` e i ǫπ z r p e i ǫπ z q` e i ℓǫπ p q p z q ` µz p p z q` µ p λ ` µ q ´ z p s p z q ` µz r p z qqq ˘ , (54) ddz ǫB ∆ q p z q ” ´p λ ` p ℓ ` q µz q ǫB ∆ p p z q ´ µ ǫB ∆ q p z q ´ ǫB ∆ s p z q` e i ǫπ ∆ q p e i ǫπ z q ` e i ℓǫπ ` µ ∆ p p z q ` ∆ q p z q ˘ ` e i ℓǫπ µ p λ ` µ q ´ ` µz ∆ r p z q ` ∆ s p z q ˘ `p ` e i ǫπ q ` p λ ` p ´ e i ǫπ qp ℓ ` q µz q p p e i ǫπ z q` µ q p e i ǫπ z q ` s p e i ǫπ z q ˘ , (55) z ddz ǫB ∆ r p z q ” p λ ` µ q ǫB ∆ p p z q ` p p ℓ ´ q ` µz q z ǫB ∆ r p z q ` ǫB ∆ s p z q` e ´ i ǫπ ∆ r p e i ǫπ z q ` e i ℓǫπ ∆ r p z q´p ` e ´ i ǫπ q ` p λ ` µ q p p e i ǫπ z q` e i ǫπ µz r p e i ǫπ z q ` s p e i ǫπ z q ˘ , (56) z ddz ǫB ∆ s p z q ” p λ ` µ q ǫB ∆ q p z q ´ p λ ` p ℓ ` q µz q z ǫB ∆ r p z q`p µ ` p ℓ ´ q z q ǫB ∆ s p z q` e ´ i ǫπ ∆ s p e i ǫπ z q ´ e i ℓǫπ pp λ ` µ q ∆ p p z q ` µz ∆ r p z qq´p ` e ´ i ǫπ q ` p λ ` µ q q p e i ǫπ z q ` µ s p e i ǫπ z q´ e i ǫπ z p λ ` p ´ e i ǫπ qp ℓ ` q µz q r p e i ǫπ z q ˘ . (57)Here ǫ P r´ , s is the auxiliary real parameter, the symbols p, q, r, s stay forarbitrary functions holomorphic in the vicinity of an arc of the circle connecting e ´ i ǫπ { with e i ǫπ { and passing inbetween them through ` counter-clockwise. The PECIAL FUNCTIONS ASSOCIATED WITH S DCHE 21 functions ǫB ∆ ✪ p z q , where ✪ “ p, q, r, s, are defined as follows. ǫB ∆ p p z q “ p p e i ǫπ z q ´ e i ǫℓπ p λ ` µ q ´ ` µz r p z q ` s p z q ˘ , ǫB ∆ q p z q “ q p e i ǫπ z q ` e i ǫℓπ `` µ z p p z q ` q p z q ˘ ` µ p λ ` µ q ´ z ` µz r p z q ` s p z q ˘˘ , ǫB ∆ r p z q “ r p e i ǫπ z q ` e i ǫℓπ r p z q , (58) ǫB ∆ s p z q “ s p e i ǫπ z q ´ e i ǫℓπ ` p λ ` µ q p p z q ` µz r p z q ˘ . (59)Lastly, the symbols ∆ ✪ p z q , where ✪ “ p, q, r, s, stand for the differences of theleft- and right-hand sides of the equations (1), (2), (3), (4), respectively. Theywere also used in Eq.s (45)-(48), (50)-(53).Thus the equalities (54)-(57) signify the pairwise coincidences, upon simplifi-cation, of certain expressions constructed in two different ways from arbitraryholomorphic functions p, q, r, s and their first order derivatives.Let us now consider the case ǫ “ . The definitions (58) are pertinent ifthe domain of functions p, q, r, s covers ˘ i and , i.e., for example, if they areholomorphic on the circular arc passing through ´ i , and ` i . The solutions ofthe Cauchy problem for Eq.s (1)-(4) with initial data specified at z “ possesssuch a property. It holds for them, by definition, ∆ ✪ p z q “ . Besides, due to theabove choice of ǫ , the summands in right-hand sides of Eq.s (54)-(57), involvingeither the factor p ` e i ǫπ q or the factor p ` e ´ i ǫπ q on the left, have to be discardedas well.Taking the above simplifications into account, Eq.s (23) follow since the argu-ment for which the transformed pqrs -functions on the left in Eq.s (17)-(20) haveto be evaluated is exactly the limit of e i ǫπ z reached as ǫ Õ (provided z belongsto the vicinity of ´ i , at least). Appendix D. Identities leading to Eq.s (24)Let us consider “the deformed differences” ǫB ∆ ✪ p z q , where ✪ “ p, q, r, s, and ǫ P r´ , s is the real parameter, defined by the formulas (58). If ǫ Õ then theyreduce to the differences B ∆ ✪ p z q of the left- and right-hand sides of Eq.s (17)-(20).We consider their values in the case z “ ´ i “ e ´ i π { . The following four doubleequalities hold true for arbitrary functions p, q, r, s holomorphic on the circulararc passing through ´ i , ` , and ` i . ǫB ∆ p p e ´ i ǫπ { q “ p p e i ǫπ { q ´ e i ǫℓπ p λ ` µ q ´ ` t r - s u µ r p e ´ i ǫπ { q ` s p e ´ i ǫπ { q ˘ ” t u p p e i ǫπ { q´ e i ǫℓπ p λ ` µ q ´ ` t r - s u µ C ∆ r p e i ǫπ { q ` C ∆ s p e i ǫπ { q ˘ , (60) ǫB ∆ q p e ´ i ǫπ { q “ q p e i ǫπ { q ` e i ǫℓπ ` t r - s u µ p p e ´ i ǫπ { q ` q p e ´ i ǫπ { q` t r - s u p λ ` µ q ´ µ p t r - s u µ r p e ´ i ǫπ { q ` s p e ´ i ǫπ { qq ˘ ” t - u ` µ p p e i ǫπ { q ´ q p e i ǫπ { q ´ r p e i ǫπ { q ˘ ` t u ` q p e i ǫπ { q ´ µ p p e i ǫπ { q ˘ ` e i ǫℓπ ` C ∆ q p e i ǫπ { q ` t r - s u µ C ∆ p p e i ǫπ { q` t r - s u p λ ` µ q ´ µ p t r - s u µ C ∆ r p e i ǫπ { q ` C ∆ s p e i ǫπ { qq ˘ , (61) ǫB ∆ r p e ´ i ǫπ { q “ r p e i ǫπ { q ` e i ǫℓπ r p e ´ i ǫπ { q” ´ ` µ p p e i ǫπ { q ´ q p e i ǫπ { q ´ r p e i ǫπ { q ˘ ` t u ` t u µ p p e i ǫπ { q ´ q p e i ǫπ { q ˘ ` e i ǫℓπ C ∆ r p e i ǫπ { q , (62) ǫB ∆ s p e ´ i ǫπ { q “ s p e i ǫπ { q ´ e i ǫℓπ ` t r - s u µ r p e ´ i ǫπ { q ` p λ ` µ q p p e ´ i ǫπ { q ˘ ” ´ t r - s u µ ` µ p p e i ǫπ { q ´ q p e i ǫπ { q ´ r p e i ǫπ { q ˘ ` t u ` t u µ q p e ´ i ǫπ { q ` s p e ´ i ǫπ { q ` t - u µ p p e ´ i ǫπ { q ˘ ´ e i ǫℓπ p λ ` µ q C ∆ p p e i ǫπ { q ´ e i ǫℓπ t r - s u µ C ∆ r p e i ǫπ { q . (63)Here we use the following auxiliary abbreviations: t - u “ e i ǫπ , t r - s u “ e ´ i ǫπ , t u “ ` e i ǫπ , t u “ ´ e i ǫπ . (64)The expressions C ∆ ✪ , where ✪ “ p, q, r, s, were introduced in the proof of Theorem2. They denote the differences of the left- and right-hand sides of Eq.s (13)-(16).In each of the above four pairs of equalities the first ones are merely the expan-sions of the corresponding definitions (58) with regard to the particular value of z picked out above. The second equalities are the identities in which the right-handsides represent some special rearrangements of the left-hand ones whose severalconstituents are aggregated to the expressions C ∆ ✪ . Thus Eq.s (60)-(63) expressthe coincidences, upon simplification, of some linear combinations of arbitraryfixed functions p, q, r, s evaluated at z “ e i ǫπ { and at z “ e ´ i ǫπ { .If ǫ “ then the argument of all the pqrs -functions and the expressions ǫB ∆ ✪ considered as the functions of z is ` . Let ǫ be further varied through the segment r , s . Then the arguments of the functions involved in Eq.s (60)-(63) move alongthe circular arcs, either clockwise of counter-clockwise. The values the functionsassume thereat can be regarded as the result of their analytic continuation fromthe vicinity of ` . At end points of the noted arcs corresponding to ǫ “ the arguments of the functions become either e i π { “ i or e ´ i π { “ ´ i whilethe expressions ǫB ∆ ✪ on the left turn into B ∆ ✪ “ lim ǫ Õ ǫB ∆ ✪ evaluated at ´ i .Besides, we also obtain t - u “ t r - s u “ ´ , t u “ , t u “ thereat. PECIAL FUNCTIONS ASSOCIATED WITH S DCHE 23
Taking these simplifications into account, one finds that in the particular caseunder the consideration the equalities of the first and the last expressions in eachof the formulas Eq.s (60)-(63) combine to Eq.s (24).
Appendix E. Identities utilized in the proof of Theorem 6
The following two identities verifiable by straightforward computation holdtrue for arbitrary holomorphic functions p, q, r, s . e ´ ǫπ Y z Ø e i ǫπ { z D u ” t D u ` e ´ i ǫℓπ z p ℓ ´ q ` ǫ ð s p { z q ǫA ∆ p p z q ´ ǫ ð r p { z q ǫA ∆ q p z q ˘ ´ ` r p z q ` µ z ´ p p z q ˘ ǫA ∆ r p z q ´ p p z q ǫA ∆ s p z q , (65) e ´ ǫπ Y z Ø e i ǫπ z D u ” t D u ` e ´ i ǫℓπ z p ´ ℓ q v e ´ i ǫℓπ p ǫ ð s ǫB ∆ p ´ ǫ ð r ǫB ∆ q q ` p µ z p ` q q ǫB ∆ r ` p λ ` µ q ´ p µ z r ` s qp µ z ǫB ∆ r ` ǫB ∆ s q w . (66)Here ǫ P r´ , s is a real variable. t D u denotes the right-hand side of Eq. (25).Eq.s (49) play role of the definition of the symbols ǫA ∆ ✪ , where ✪ “ p, q, r, s. Similarly, Eq.s (58) explain the meaning of the symbols ǫB ∆ ✪ . The renderings ofall these abbreviations, as they stand, are considered as the functions of z . Herewe also employ in recording some tricks enabling us to produce somewhat morecompact presentations than in preceding Appendices. In particular, ‘the accent’ ǫ ð denotes the transformation of rotation of the function argument at an angle ǫπ , i.e. ǫ ð ✪ p z q “ ✪ p e i ǫπ z q . Note that the arguments of functions are displayed inEq. (65) (except for t D u ) but they are suppressed in (66) because in the lattercase the arguments of all the functions coincide and are equal to z .To guarantee the meaningfulness of the formulas Eq.s (65) and (66) one has toensure the belonging of the values of arguments, for which the functions involvedin them are evaluated, to the appropriate domain. These values depend on ǫ . Inparticular, if ǫ “ then all the functions are evaluated either at z or at { z . Insuch a case one may get any z P C ˚ for which both formulas (65), (66) proveto be correctly defined — and the equalities they represent hold true. Further,starting with ǫ “ , we carry out analytic continuations of all the constituentsof Eq.s (65) and (66) varying ǫ P r , s from 0 to 1. In the limit ǫ Õ (i.e. atthe end point of the arc of analytic continuation) the expressions denoted ǫA ∆ ✪ and ǫB ∆ ✪ become identical to the expressions A ∆ ✪ and B ∆ ✪ , respectively (see thediscussions following Eq. (30) and Eq. (32)), while the transformation indicatedby ‘the accent’ ǫ ð converts to the semi-monodromy transformation denoted earlierby ‘the accent’ ð (and equivalent to action of the operator M { ).Inspecting the result of the outlined analytic continuation along the image ofthe segment r , s P ǫ , one finds that this is nothing else but the equations (30)and (32), provided that z belongs to the vicinity of ` i in the former case and Im z ă in the latter one. Appendix F. Identities utilized in the proof of Theorem 7
The following identity, which is verifiable by straightforward computation,holds true for arbitrary holomorphic functions
E, p, q, r, s . Y z Ø e i ǫπ { z e ´ µ p z ` { z q p H ˝ ǫ L A qr E s \ ” z p H r E s` t r - s u ` µ p t - ut u ` t u z q p ´ t - u q ` t - u z r ˘ H r E s` z E ∆ p ` z E ∆ q ` t r - s u ` t - u p λ ´ p ℓ ` q µz q E ` t - u z E ` p µ p t ut u ´ ` z q ` t - u p µ ` z qq E ˘ ∆ p ` t r - s u `` µ p t ut u ´ ` z q ´ t - u µ p ℓ ´ ´ µz q ˘ E ` t - u z E ˘ ∆ q ` z E ∆ r ` E ∆ s ´ t u µ ` p µ ` t r - s u z p ℓ ´ ´ µz qqp z pE ` qE q` p ´ t r - s u z qp z rE ` sE q ˘ . (67)Here ǫ P r´ , s is the auxiliary real parameter. The abbreviations t - u , t r - s u , t u , t u are to be expanded in accordance with formulas (64). The operator H is definedby Eq. (37) (see also Eq. (33)), H “ d { dz ˝ H , the operator ǫ L A is defined byEq. (34). The symbols ∆ ✪ p z q , where ✪ “ p, q, r, s, denote the differences of theleft- and right-hand sides of Eq.s (1), (2), (3), (4), respectively, considered as thefunctions of z .Similar identity describing this time the composition of H with the operator ǫ L B (see Eq. (35)) reads z ℓ ´ e ´ µ p z ` { z q p H ˝ ǫ L B qr E s ” z ǫ ð r ǫ ð H r E s ` ` p ℓ ´ q z ǫ ð r ` ǫ ð s ` z ǫ ð r ˘ ǫ ð H r E s` z ǫ ð E ǫ ð ∆ r ` ǫ ð E ǫ ð ∆ s ´ p λ ` µ q ` ǫ ð E ǫ ð ∆ p ` ǫ ð E ǫ ð ∆ q ˘ ´ ` p µ ´ p ℓ ´ q z q ǫ ð E ` p λ ` p ℓ ` q µz q ǫ ð E ˘ ǫ ð ∆ r ` p ǫ ð E ´ µ ǫ ð E q ǫ ð ∆ s ` t u ` W ǫ ð E ` W ǫ ð E ´ W ǫ ð E ´ W ǫ ð E ˘ , (68)where the following abbreviations are employed W “ p ℓ ` qp λ ` µ q µz ǫ ð p ` z ` ´ p ℓ ´ q λ ` p ℓ ` q µz p ´ t u ` t - u µz q ˘ ǫ ð r ` µz ǫ ð s ´ z ` t u λ ´ t - u p ℓ ` qp ´ t u q µz ˘ ǫ ð r ´ µ p ´ t - u z q ǫ ð s ,W “ ´p ℓ ´ qp λ ` µ q z ǫ ð p ` z ` ´ µ p ℓ ´ t u q ` pp ℓ ´ qp ´ t u q ´ t u λ q z ` t - ut u p ℓ ` q µz ˘ ǫ ð r PECIAL FUNCTIONS ASSOCIATED WITH S DCHE 25 ´ ` p ℓ ´ q z ` µ p ´ t - u z q ˘ ǫ ð s ´ t u p λ ` µ q z ǫ ð p ´ z ` t - u p ` t - u p ℓ ´ qq z ` µ p t u ` t - u p ´ z qq ˘ ǫ ð r ´ t u z ǫ ð s ´ t ut - u z ǫ ð r ,W “ t ut - u z ǫ ð r ` z ` t - u p ` t u p ℓ ´ qq z ` µ p t u ` t - u p ´ z qq ˘ ǫ ð r ,W “ t ut - u z ǫ ð r . The meaning of the ∆ -symbols and the abbreviations t - u , t u , t u , was explainedabove. ‘The diacritic mark’ ǫ ð , used also in Eq. (66), denotes the transformationcarrying out the rotation of the function argument at an angle ǫπ .If ǫ “ then the arguments of all the functions involved in Eq.s (67) and (68)are either z or (somewhere in (67)) { z . Accordingly, for any z P C ˚ all theconstituents of the both formulas (67), (68) are well defined — and the equali-ties they signify hold true for arbitrary functions E, p, q, r, s holomorphic in C ˚ .Further, we allow ǫ to vary through the segment r , s and carry out analyticcontinuation along the corresponding curves (in fact, the circular arcs) of all theconstituents of the formulas (67), (68). Fixing the result of this analytic contin-uation at the end points corresponding to ǫ “ , we obtain the two equalities inwhich some noted abbreviations acquire the known numerical values as follows: t - u “ t r - s u “ ´ , t u “ , t u “ . Then it is easy to see that we obtain in thisway the equations (in fact, identities) (38) and (39).To prevent egresses of the points of evaluation of our functions from C ˚ , itis enough to pick z from the vicinity of ` i in the case of Eq. (38) and from thehalf-plane Im z ă in the case of Eq. (39). The extending to greater domainscan be ensured by means of an analytic continuation. Appendix G. Identities utilized in the proof of Theorem 8
The identities given below represent the appropriately adapted expansions ofthe iterated linear operators ǫ L A and ǫ L B defined by Eq.s (34), (35). Specifically,let E, p, q, r, s denote arbitrary holomorphic functions and ǫ P r´ , s be the realparameter. Then it can be shown by means of straightforward computationsshow that, at first, p ǫ L A ˝ ǫ L A qr E sp z q ` e i ℓπ t D u E p z q ” ` p s p z q ` µ z ´ q p z qq E p z q ` p z r p z q ` µ p p z qq E p z q ˘ ǫA ∆ p p z q` ` q p z q E p z q ` z p p z q E p z q ˘ ǫA ∆ q p z q` ǫ ð p p { z q ` p p z q H r E sp z q ` E p z q ∆ p p z q ` E p z q ∆ q p z q ˘ ` t u ǫ ð p p { z q W ` ` e µ p t r s u z ` t u { z q ´ ˘ W , (69) where W “ ` t u µz ´ q p z q ` q p z q ˘ E p z q ` z p p z q E p z q` ` q p z q ` p t u µ ` z q p p z q ` z p p z q ˘ E p z q ,W “ ´ ǫ ð p p { z q ´` µz ´ p z ´ t - u q q p z q ` t - u q p z q ˘ E p z q` ` p µ p z ´ t - u q ` t - u z q p p z q` t - u q p z q ` t - u z p p z q ˘ E p z q` t - u z p p z q E p z q ¯ ` ǫ ð q p { z q ` q p z q E p z q ` z p p z q E p z q ˘ and, at second, e i ℓǫπ z p ℓ ´ q v p ǫ L B ˝ ǫ L B qr E s ` e ℓπ p λ ` µ q ǫ ð t D u ǫ ð E w ”´ t - u z ǫ ð E ¨ ` p λ ` µ q ǫ ð r ǫ ð B ∆ p ` p µ ǫ ð z ǫ ð r ` ǫ ð s q ǫ ð B ∆ r ˘ ´ ǫ ð E ¨ pp λ ` µ q ǫ ð r ǫ ð B ∆ q ` p µ ǫ ð z ǫ ð r ` ǫ ð s q ǫ ð B ∆ s q` t - u z ǫ ð r ǫ ð r ǫ ð H r E s ` t - u z ǫ ð r ǫ ð E ǫ ð ∆ r ` ǫ ð r ǫ ð E ǫ ð ∆ s ´ t u W ` p e t u µ p z ` t - u { z q ´ q t - u W . (70)where W “ p µ p ´ ǫ ð z q ǫ ð r ´ ǫ ð s q ` t - u z ǫ ð r ǫ ð E ` ǫ ð s ǫ ð E ˘ ,W “ ` ǫ ð s ǫ ð s ` ǫ ð r ¨ ` ´p µ p ´ t - u z q ` t - u p ℓ ´ q z q ǫ ð s ` t - u z ǫ ð s ˘˘ ǫ ð E ` z t - u ´ t - u ǫ ð s ǫ ð r ǫ ð E ` ǫ ð r ¨ ` p ǫ ð s ` t - u z ǫ ð r q ǫ ð E ` t - u ǫ ð r ¨ p´p µ p ´ t - u z q ` t - u p ℓ ´ q z q ǫ ð E ` t - u z ǫ ð E q ˘¯ . Here we employ, in particular, the following abbreviations t - u “ e i ǫπ , t u “ ` e i ǫπ , t r s u “ ` e ´ i ǫπ , t u “ ´ e i ǫπ . (71)Some further abbreviations are also used for convenience: t D u stands for the right-hand side of Eq. (25) considered as a function of z . The operator H is definedby Eq. (37). The symbols ∆ ✪ p z q , where ✪ “ p, q, r, s, denote the differencesof the left- and right-hand sides of Eq.s (1), (2), (3), (4), respectively, whichare considered as the functions of z . Similarly, the symbols ǫA ∆ ✪ stand for thecorresponding ‘ ǫ -deformed’ differences A ∆ ✪ of the left- and right-hand sides ofEq.s (6)-(9) which are defined by Eq.s (49). ‘The diacritic mark’ ǫ ð indicates thetransformation carrying out the rotation of the function argument at an angle ǫπ , i.e. ǫ ð ✪ p z q “ ✪ p e i ǫπ z q . Similar ‘accent’ ǫ ð carries the concordant meaning: ascompared to ǫ ð , the rotation angle for it amounts to ǫπ . PECIAL FUNCTIONS ASSOCIATED WITH S DCHE 27
Certain abuse of notations is related to the symbols ǫ ð B ∆ ✪ . Similarly to thesymbols ǫA ∆ ✪ , they refer to the ‘ ǫ -deformed’ differences of the left- and right-sides of (this time) Eq.s (17)-(20), which are defined by Eq.s (58). However, nowadditionally ‘the ǫ ð -rotation’ of argument of the result considered as a functionof z has to be carried out afterwards. Thus in this case one deals, in a sense,with the ‘ ǫ -deformed and ǫ ð -rotated’ differences of the left- and right-hand sidesof Eq.s (17)-(20).If ǫ “ then all the instances of the functions E, p, q, r, s and their derivativesinvolved in Eq.s (69), (70) are evaluated either at z or at { z . Then all theconstituents of the formulas (69) and (70) are well defined for arbitrary z P C ˚ — and the equalities they represent hold true.Next, we allow the parameter ǫ to vary through the segment r , s and carryout analytic continuation along the corresponding curves (in fact, circular arcs) inthe function domains. At their end points corresponding to ǫ “ the coefficientsrepresented by the abbreviations t - u , t r - s u , t u , t u acquire the values ´ , ´ , , ,respectively, see Eq.s (71). This allows us in particular to ignore the last linesin the both formulas (69) and (70) becoming null. Simultaneously, the effect ofthe rotation of the function arguments tagged by ‘the accent’ ǫ ð turns into theaction of the semi-monodromy operator M { (see Theorem 3) which we indicatealso by ‘the accent’ ð over the function symbol, see, e.g., the proof of Theorem8. As to the exceptional symbols ǫ ð B ∆ ✪ , it is easy to see that at the end point ofthe curve of analytic continuation they become equal to the expressions denotedin Eq. (44) by the symbols ð B ∆ ✪ , ✪ “ p, q, r, s .A separate note on the effect of ‘the accent’ ǫ ð is necessary. In the limit as ǫ Õ it also ‘rotates’ the argument of the function to be transformed but nowthe corresponding angle amounts to π meaning, in a sense, a full revolution. Ithad been noticed that such transformations are termed monodromy. We denotedthe operator carrying out the monodromy transformation by the symbol M butin some formulas (e.g. in Eq. (44)) it is also indicated by ‘the diacritic mark’ ö .It has also to be noted that in case of monodromy transformation some pre-caution on structure of the domain of the function to which it acts needs to betaken. This point is briefly discussed in the proof of Theorem 8.Now, collecting all the modifications of the formulas Eq.s (69) and (70), arisingwhen the analytic continuation corresponding to lim ǫ Õ1