aa r X i v : . [ m a t h . C A ] J a n Dihedral Gauss hypergeometric functions
Raimundas Vidunas
Kobe University
June 2, 2018
Abstract
Gauss hypergeometric functions with a dihedral monodromy group can be ex-pressed as elementary functions, since their hypergeometric equations can be trans-formed to Fuchsian equations with cyclic monodromy groups by a quadratic changeof the argument variable. The paper presents general elementary expressions ofthese dihedral hypergeometric functions, involving finite bivariate sums expressibleas terminating Appell’s F or F series. Additionally, trigonometric expressions forthe dihedral functions are presented, and degenerate cases (logarithmic, or with themonodromy group Z / Z ) are considered. As well known, special cases of the Gauss hypergeometric function F (cid:16) A, BC (cid:12)(cid:12)(cid:12) z (cid:17) canbe represented in terms of elementary functions. A particularly interesting case arehypergeometric functions with a dihedral monodromy group; they can be expressed withsquare roots inside power or logarithmic functions. The simplest examples are: F (cid:18) a , a +12 a + 1 (cid:12)(cid:12)(cid:12)(cid:12) z (cid:19) = (cid:18) √ − z (cid:19) − a , (1.1) F a , a +1212 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z ! = (1 − √ z ) − a + (1 + √ z ) − a , (1.2) F a +12 , a +2232 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z ! = (1 − √ z ) − a − (1 + √ z ) − a a √ z ( a = 0) , (1.3) F , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z ! = log(1 + √ z ) − log(1 − √ z )2 √ z = arctan √− z √− z , (1.4) F , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z ! = log( √ − z + √− z ) √− z = arcsin √ z √ z . (1.5)These are solutions of the hypergeometric differential equation with the local exponentdifferences 1 / , / , a at the three singular points. The monodromy group is an infinite1ihedral group (for general a ∈ C ), or a finite dihedral group (for rational non-integer a ), or an order 2 group (for non-zero integers a ). We refer to Gauss hypergeometricfunctions with a dihedral monodromy group as dihedral hypergeometric functions .Despite a rich history of research of Gauss hypergeometric functions, only simplestformulas for dihedral F functions like above are given in common literature [1, 15.1], [4,2.8]. General dihedral F functions are contiguous to the simplest functions given above(or their fractional-linear transformations); that is, their upper and lower parametersdiffer by integers from the respective parameters of the simplest functions. The localexponent differences of their hypergeometric equations are k + 1 / ℓ + 1 / λ ∈ C ,where k, ℓ are integers.An interesting problem is to find explicit elementary expressions for general dihedralGauss hypergeometric functions. A set of these expressions is presented in Sections 3,4, 6 of this paper. The most general canonical expressions are given in Section 3, interms of terminating Appell’s F double sums. The key observation is that a particularunivariate specialization of Appell’s F function satisfies the same Fuchsian equation asa quadratic transformation of general hypergeometric equations; see Theorem 2.1. Whenthe monodromy group of the hypergeometric equation is dihedral, the alluded F functionis a terminating double sum. Linear relations between dihedral F and terminating F solutions of the same Fuchsian equation give the announced elementary expressions forthe former.Section 4 looks at the simpler case of dihedral hypergeometric equations with the localmonodromy differences k + 1 / , / , λ . Then the terminating F doublesums becometerminating F sums. The case of the local exponent differences k + 1 / , k + 1 / , λ canbe reduced to the described case by a quadratic transformation of the hypergeometricequation.Section 5 considers comprehensively degenerate cases with λ ∈ Z .Section 6 considers trigonometric formulas for dihedral Gauss hypergeometric func-tions. The simplest such expressions are [1, 15.1], [4, 2.8]: F a , − a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin x ! = cos ax, (1.6) F a , − a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin x ! = sin axa sin x . (1.7)Supplementing paper [10] presents similar expressions (as double hypergeometricsums) for quadratic invariants for hypergeometric equations with the dihedral mon-odromy group, and describes rational pull-back transformations between dihedral hy-pergeometric functions. 2 Preliminary facts
Basic facts on hypergeometric functions, Fuchsian equations, the monodromy group,contiguous relations, terminating hypergeometric sums, Zeilberger’s algorithm are wellknown. We suggest [2], [3], [5] as standard though overwhelming references. Wikipediapages [11] can be satisfactorily consulted for quick references.
The Gauss hypergeometric function F (cid:18) A, BC (cid:12)(cid:12)(cid:12)(cid:12) z (cid:19) satisfies the hypergeometric differen-tial equation [2, Formula (2.3.5)]: z (1 − z ) d y ( z ) dz + (cid:0) C − ( A + B + 1) z (cid:1) dy ( z ) dz − A B y ( z ) = 0 . (2.1)This is a canonical Fuchsian equation on P with three singular points. The singularitiesare z = 0 , , ∞ , and the local exponents are:0, 1 − C at z = 0; 0, C − A − B at z = 1; and A , B at z = ∞ .The local exponent differences at the singular points are equal (up to a sign) to 1 − C , C − A − B and A − B , respectively. Let us denote a hypergeometric equation with thelocal exponent differences d , d , d by E ( d , d , d ), and consider the order of the threearguments unimportant.Because of frequent use, we recall Euler’s and Pfaff’s fractional-linear transformations[2, Theorem 2.2.5]: F (cid:18) a, bc (cid:12)(cid:12)(cid:12)(cid:12) z (cid:19) = (1 − z ) c − a − b F (cid:18) c − a, c − bc (cid:12)(cid:12)(cid:12)(cid:12) z (cid:19) (2.2)= (1 − z ) − a F (cid:18) a, c − bc (cid:12)(cid:12)(cid:12)(cid:12) zz − (cid:19) . (2.3)The following quadratic transformation [2, (3.1.3),(3.1.9),(3.1.7)] will illustrate thekey reduction of the dihedral monodromy group to a cyclic monodromy group: F (cid:18) a, b a + b +12 (cid:12)(cid:12)(cid:12)(cid:12) x (cid:19) = F a , b a + b +12 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x (1 − x ) ! , (2.4) F (cid:18) a, ba − b + 1 (cid:12)(cid:12)(cid:12)(cid:12) x (cid:19) = (1 + x ) − a F (cid:18) a , a +12 a − b + 1 (cid:12)(cid:12)(cid:12)(cid:12) x (1 + x ) (cid:19) , (2.5) F (cid:18) a, b b (cid:12)(cid:12)(cid:12)(cid:12) x (cid:19) = (cid:16) − x (cid:17) − a F a , a +12 b + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x (2 − x ) ! . (2.6)The hypergeometric equations are related in the same way, by a quadratic pull-backtransformation of the form z ϕ ( x ) , y ( z ) Y ( x ) = θ ( x ) y ( ϕ ( x )) , (2.7)3here ϕ ( x ) is a rational (quadratic in this case) function, and θ ( x ) is a power factor.Geometrically, the transformation pull-backs the starting differential equation on theprojective line P z to a differential equation on the projective line P x , with respect tothe covering ϕ : P x → P z . The factor θ ( x ) shifts the local exponents of the pull-backedequation, but it does not change the local exponent differences. We write the quadratictransformation of hypergeometric equations as E (cid:0) , λ, µ (cid:1) ←− E ( λ, λ, µ ) , (2.8)as the local exponent differences 1 / , µ are doubled by the quadratic substitution, and thepoint with the local exponent difference 1 / θ ( x ). The arrow follows the direction of the pull-back covering, as in [8], [10]. The simplest hypergeometric equations with a dihedral monodromy group are E (1 / , / , a ).The monodromy representation for these equations group can be computed using explicitexpressions (1.1)–(1.4). If a = 0, we take (1.2) and a √ z times (1.3) as a basis of solutions;analytic continuation along loops around z = 0 and z = 1 gives the following generatorsof the monodromy group: (cid:18) − (cid:19) and (cid:18) w − w − w w (cid:19) , w = exp( − πia ) . (2.9)If a = 0, the monodromy generators are (cid:18) − (cid:19) and (cid:18) πi (cid:19) .In general, dihedral hypergeometric functions are contiguous to E (1 / , / , a ). Theyare characterized by the property that their differences of local exponents at two of thethree singular points are half-integers. If the third local exponent is an irrational number,the monodromy group is an infinite dihedral group; if it is a non-integer rational number,the monodromy group is a finite dihedral group. If the third local exponent differenceis an integer, the monodromy group is isomorphic either to Z / Z or (in presence oflogarithmic solutions) to an infinite dihedral group. The distinction of these two cases isgiven in Theorem 5.1 below.The main results of this paper are stated for solutions of hypergeometric equation(2.1) with A = a , B = a + 12 + ℓ, C = 12 − k. (2.10)The local exponent differences of our working hypergeometric equation are k +1 / ℓ +1 / λ = a + k + ℓ . Throughout the paper, k, ℓ, m denote non-negative integers. Exceptin Section 5, we assume that a is not an integer. If we apply a quadratic pull-back transformation to a hypergeometric equation, andthe two ramified points are singularities of the equation, the transformed equation is4 Fuchsian equation with generally 4 singular points (and can be solved in terms of
Heun functions ). Suppose that the hypergeometric equation has a dihedral monodromygroup, hence it is E (cid:0) k + , ℓ + , λ (cid:1) , and the two ramified points are the points withthe half-integer local exponent differences. Then the local exponent differences of thetransformed equation are 2 k + 1 , ℓ + 1 , λ, λ . With an appropriate choice of the powerfactor θ ( x ) in (2.7), the transformed equation has trivial monodromy around the twopoints with the integer local exponent differences 2 k + 1, 2 ℓ + 1. These two points are notlogarithmic because the corresponding points with half-integer local exponent differencesare not logarithmic. The monodromy action for the transformed equation will come onlyfrom the other two points. The global monodromy group is therefore cyclic, and themonodromy representation is reducible.In the simplest case k = 0, ℓ = 0 the transformed equation has just two singularities.Correspondingly, the classical quadratic transformation (2.4) gives F (cid:18) a , a +12 a + 1 (cid:12)(cid:12)(cid:12)(cid:12) x (1 − x ) (cid:19) = F (cid:18) a, a + 1 a + 1 (cid:12)(cid:12)(cid:12)(cid:12) x (cid:19) = (1 − x ) − a . (2.11)This formula is equivalent to (1.1). If exactly one of k, ℓ is zero, the transformed equationis equivalent to a hypergeometric equation again. The transformation is then E (cid:0) , k + , λ (cid:1) ←− E (2 k + 1 , λ, λ ) , (2.12)and the transformed solutions can be expressed in terms of terminating F sums, as wedemonstrate in Section 4.Even with general integer k , ℓ , the transformed solution must have elementary power(at worst logarithmic) solutions, because of the reducible monodromy group. With aproper normalization by θ ( x ) in (2.7) the elementary power solutions can be polynomialsin x . It turns out that those polynomials can be written as terminating Appell’s F or F hypergeometric sums. Recall that Appell’s F and F bivariate functions are definedby the series F (cid:18) a ; b , b c , c (cid:12)(cid:12)(cid:12)(cid:12) x, y (cid:19) = ∞ X p =0 ∞ X q =0 ( a ) p + q ( b ) p ( b ) q ( c ) p ( c ) q p ! q ! x p y q , (2.13) F (cid:18) a , a ; b , b c (cid:12)(cid:12)(cid:12)(cid:12) x, y (cid:19) = ∞ X i =0 ∞ X j =0 ( a ) p ( a ) q ( b ) p ( b ) q ( c ) p + q p ! q ! x p y q . (2.14)As usual, ( a ) n denotes the Pochhammer symbol (also called raising factorial ), which isthe product a ( a + 1) · · · ( a + n − F or F functions. In particular, the following is proved in [9]. Theorem 2.1.
The functions F (cid:18) a ; b , b b , b (cid:12)(cid:12)(cid:12)(cid:12) x, − x (cid:19) and ( x − − a F a , a +12 − b b + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x (2 − x ) ! (2.15)5 atisfy the same second order Fuchsian equation.Proof. Part 1 of Theorem 2.4 in [9].For general values of the 3 parameters a, b , b , the double series for the F ( x, − x )function does not converge for any x . However, when b and b are zero or negativeintegers, the F ( x, − x ) function can be seen as a finite sum of (1 − b )(1 − b ) terms;see Remark 2.3 below. On the other hand, the F function in (2.15) is contiguous tothe F function in (1.2) for integer values of b , b , hence in general it has a dihedralmonodromy group as well. This relation between terminating F ( x, − x ) sums anddihedral hypergeometric functions is behind our explicit expressions for general dihedralfunctions. We formulate the following variation of Theorem 2.1. Corollary 2.2.
For non-positive integers k , ℓ , the functions F a , a +12 + ℓ − k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z ! and (1 + √ z ) − a F (cid:18) a ; − k, − ℓ − k, − ℓ (cid:12)(cid:12)(cid:12)(cid:12) √ z √ z ,
21 + √ z (cid:19) (2.16) satisfy the same second order Fuchsian equation.Proof. Substitute b = − k , b = − ℓ and x = 2 √ z/ (1 + √ z ) in Theorem 2.1.We present explicit consequences of this coincidence of differential equations in Section3. Note that if k = 0 or ℓ = 0 then the double F ( x, − x ) sum becomes a terminating F ( x ) sum, in agreement with an observation after formula (2.11).Appell’s F ( x, y ) and F ( x, y ) functions are closely related. They satisfy the samesystem of partial differential equations up to a simple transformation, and particularly,terminating F sums become terminating F sums when summation is reversed in bothdirections. In particular, for ( a ) k + ℓ = 0 we have F (cid:18) a ; − k, − ℓ − k, − ℓ (cid:12)(cid:12)(cid:12)(cid:12) x, y (cid:19) = k ! ℓ ! ( a ) k + ℓ (2 k )! (2 ℓ )! x k y ℓ F (cid:18) k + 1 , ℓ + 1; − k, − ℓ − a − k − ℓ (cid:12)(cid:12)(cid:12)(cid:12) x , y (cid:19) . (2.17) Remark 2.3.
The hypergeometric series F (cid:18) − k, a − k (cid:12)(cid:12)(cid:12)(cid:12) x (cid:19) is not conventionally definedfor a non-negative integer k , because of the zero or negative lower parameter. But it canbe usefully interpreted in two ways: as a terminating sum of k + 1 hypergeometric terms,or by taking the term-wise limit with k ∈ R approaching a non-negative positive integer.With both interpretations, the F function is a solution of the respective hypergeometricequation.In this paper, we adopt the terminating sum interpretation for such F functions andsimilar bivariate hypergeometric sums. In particular, the F function in (2.16) is a sumof ( k + 1)( ℓ + 1) hypergeometric terms. 6 Explicit expressions for dihedral functions
The following theorem presents generalizations of (1.1)–(1.3). The identities are finiteelementary expressions for general dihedral hypergeometric functions. The F and F series are finite sums of ( k + 1)( ℓ + 1) terms. Because these formulas express solutionsof any dihedral hypergeometric equation, we refer to them as canonical.Note that the F sum in (3.1) terminates for all (positive or negative) integers k, ℓ ,as the set of upper parameters does not change under the substitutions k
7→ − k − ℓ
7→ − ℓ − Theorem 3.1.
The following formulas hold for non-negative integers k, ℓ and general a ∈ C : F (cid:18) a , a +12 + ℓa + k + ℓ + 1 (cid:12)(cid:12)(cid:12)(cid:12) − z (cid:19) = z k/ (cid:18) √ z (cid:19) − a − k − ℓ × F (cid:18) k + 1 , ℓ + 1; − k, − ℓa + k + ℓ + 1 (cid:12)(cid:12)(cid:12)(cid:12) √ z − √ z , − √ z (cid:19) , (3.1) (cid:0) a +12 (cid:1) ℓ (cid:0) (cid:1) ℓ F a , a +12 + ℓ − k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z ! = (1 + √ z ) − a F (cid:18) a ; − k, − ℓ − k, − ℓ (cid:12)(cid:12)(cid:12)(cid:12) √ z √ z ,
21 + √ z (cid:19) + (1 − √ z ) − a F (cid:18) a ; − k, − ℓ − k, − ℓ (cid:12)(cid:12)(cid:12)(cid:12) √ z √ z − , − √ z (cid:19) , (3.2) (cid:0) a +12 (cid:1) k (cid:0) a (cid:1) k + ℓ +1 (cid:0) (cid:1) k (cid:0) (cid:1) k +1 (cid:0) (cid:1) ℓ ( − k z k + F a +12 + k, a + k + ℓ + 1 + k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z ! = (1 − √ z ) − a F (cid:18) a ; − k, − ℓ − k, − ℓ (cid:12)(cid:12)(cid:12)(cid:12) √ z √ z − , − √ z (cid:19) − (1 + √ z ) − a F (cid:18) a ; − k, − ℓ − k, − ℓ (cid:12)(cid:12)(cid:12)(cid:12) √ z √ z ,
21 + √ z (cid:19) . (3.3) Proof.
The first identity follows directly from Karlsson’s identity [6, 9.4.(90)]; the serieson both sides coincide in a neighborhood of z = 1. Other local solution at z = 1 is(1 − z ) − a − k − ℓ F (cid:18) − a − k − ℓ, − a − k − a − k − ℓ (cid:12)(cid:12)(cid:12)(cid:12) − z (cid:19) , (3.4)which evaluates to z k/ (cid:0) − √ z (cid:1) − a − k − ℓ F (cid:18) k + 1 , ℓ + 1; − k, − ℓ − a − k − ℓ (cid:12)(cid:12)(cid:12)(cid:12) √ z − √ z , − √ z (cid:19) (3.5)by (3.1) with the substitution a
7→ − a − k − ℓ .The three terms in (3.2) are solutions of the same second order Fuchsian equation byCorollary 2.2, so there must be a linear relation between them. Up to a scalar multiple,the right-hand side of (3.2) is the only linear combination of the two F terms whichis invariant under the conjugation √ z
7→ −√ z . Hence the left and right-hand sides of73.2) differ by a factor independent of z . Evaluation of the right side at z = 0 leads to aterminating F (2) sum. It remains to prove that F (cid:18) a, − ℓ − ℓ (cid:12)(cid:12)(cid:12)(cid:12) (cid:19) = (cid:0) a +12 (cid:1) ℓ (cid:0) (cid:1) ℓ . (3.6)Zeilberger’s algorithm is by now a routine technique to find two-term hypergeometricidentities or recurrence relations for hypergeometric sums [5], [2, Section 3.11]. Its wholeoutput includes certificate information, that allows to check the identity or recurrencerelation without computer assistance. Particularly, Zeilberger’s algorithm returns thefollowing difference equation for the F (2) summand S ( ℓ, j ):(2 ℓ + 1) S ( ℓ + 1 , j ) − ( a + 1 + 2 ℓ ) S ( ℓ, j ) = H ( ℓ, j + 1) − H ( ℓ, j ) , (3.7)where S ( ℓ, j ) = ( a ) j ( − ℓ ) j j ! ( − ℓ ) j j , H ( ℓ, j ) = − j (2 ℓ + 1 − j )2 ( ℓ + 1 − j ) S ( ℓ, j ) . This certificate identity can be checked by hand straightforwardly. To avoid the 0 / j = ℓ + 1, we may also write H ( ℓ, j ) = (1 − a − j ) S ( ℓ, j − j = 0 , , . . . , ℓ + 1 we obtain a telescoping sum on theright hand side that simplifies to H ( ℓ, ℓ + 2) − H ( ℓ,
0) = 0. The summation of the left-hand side gives a first order difference equation for the F (2) sum. The same differenceequation is satisfied by the right-hand side of (3.6), and a check that the initial values at ℓ = 0 coincide completes the proof of (3.6).As an intermediate step between formulas (3.1)–(3.2) and (3.3), we derive formula(3.8) in Lemma 3.2 immediately below. Then we use connection formula [2, (2.3.13)],with both right-side terms transformed by Euler’s transformation (2.2): z k + F a +12 + k, a + k + ℓ + 1 + k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z ! = Γ (cid:0) + k (cid:1) Γ( − a − k − ℓ )Γ (cid:0) − a (cid:1) Γ (cid:0) − a − ℓ (cid:1) F (cid:18) a , a +12 + ℓa + k + ℓ + 1 (cid:12)(cid:12)(cid:12)(cid:12) − z (cid:19) + Γ (cid:0) + k (cid:1) Γ( a + k + ℓ )Γ (cid:0) a +12 + k (cid:1) Γ (cid:0) a + k + ℓ + 1 (cid:1) (1 − z ) − a − k − ℓ F (cid:18) − a − k − ℓ, − a − k − a − k − ℓ (cid:12)(cid:12)(cid:12)(cid:12) − z (cid:19) . After applying evaluation (3.1) to both right-side terms and using relations (2.17), (3.8),we get (3.3).
Lemma 3.2.
We have the following transformation formulas for the terminating F and sums appearing in (3.1)–(3.3) : (cid:0) √ z (cid:1) k + ℓ F (cid:18) a ; − k, − ℓ − k, − ℓ (cid:12)(cid:12)(cid:12)(cid:12) √ z √ z ,
21 + √ z (cid:19) =( − ℓ (cid:0) a +12 (cid:1) ℓ (cid:0) a +12 + k (cid:1) ℓ (cid:0) − √ z (cid:1) k + ℓ F (cid:18) − a − k − ℓ ; − k, − ℓ − k, − ℓ (cid:12)(cid:12)(cid:12)(cid:12) √ z √ z − , − √ z (cid:19) . (3.8) F (cid:18) k + 1 , ℓ + 1; − k, − ℓa + k + ℓ + 1 (cid:12)(cid:12)(cid:12)(cid:12) √ z − √ z , − √ z (cid:19) =( a ) k + ℓ (cid:0) a +12 + k (cid:1) ℓ (1 + a + k + ℓ ) k + ℓ (cid:0) a +12 (cid:1) ℓ F (cid:18) k + 1 , ℓ + 1; − k, − ℓ − a − k − ℓ (cid:12)(cid:12)(cid:12)(cid:12) √ z + 12 √ z , √ z (cid:19) . (3.9) Proof.
Connection formula [2, (2.3.13)] gives F a , a +12 + ℓ − k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z ! = Γ (cid:0) − k (cid:1) Γ( − a − k − ℓ )Γ (cid:0) − a − k (cid:1) Γ (cid:0) − a − k − ℓ (cid:1) F (cid:18) a , a +12 + ℓa + k + ℓ + 1 (cid:12)(cid:12)(cid:12)(cid:12) − z (cid:19) + Γ (cid:0) − k (cid:1) Γ( a + k + ℓ )Γ (cid:0) a (cid:1) Γ (cid:0) a +12 + ℓ (cid:1) (1 − z ) − a − k − ℓ F (cid:18) − a − k − ℓ, − a − k − a − k − ℓ (cid:12)(cid:12)(cid:12)(cid:12) − z (cid:19) . We evaluate both terms on the right-hand side using (3.1), apply (2.17) twice and comparethe whole formula with (3.2). The functions (1 + √ z ) − a and (1 − √ z ) − a are algebraicallyindependent in general, and identification of respective terms to them gives (3.8).Formula (3.9) follows from (3.8) via (2.17).Summarising, the F functions (3.2) and (3.3) form a local basis of solutions at x = 0;the functions in (3.1) and (3.4) form a local basis of solutions at x = 1; the functions z − a F a , a +12 + k − ℓ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z ! , z − a +12 − ℓ F a +12 + ℓ, a + k + ℓ + 1 + ℓ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z ! (3.10)form a local basis of solutions at x = ∞ . Each of the six functions can be (generally)expressed by a hypergeometric series in 4 different ways following Euler-Pfaff transfor-mations (2.2)–(2.3). This gives the standard set (and structure) of 6 × • Permutation of the local exponents at z = 0: the parameters are transformed as k
7→ − k − a a + 2 k + 1; the hypergeometric solution gets multiplied by z k +1 / . • Permutation of the local exponents at z = 1: the parameters are transformed as a
7→ − a − k − ℓ ; the hypergeometric solution gets multiplied by (1 − z ) − a − k − ℓ . • Permutation of the local exponents at z = ∞ : the parameters are transformed as ℓ
7→ − ℓ − a a +2 ℓ +1; the hypergeometric solution gets multiplied by z − ℓ − / .9 Permutation z /z of the singularities z = 0, z = ∞ : the parameters aretransformed as k ↔ ℓ ; the hypergeometric solution gets multiplied by z − a/ . • Permutation z − z of the singularities z = 0, z = 1: the parameters aretransformed as k
7→ − a − k − ℓ − . • Permutation z z/ ( z −
1) of the singularities z = 1, z = ∞ : the parameters aretransformed as ℓ
7→ − a − k − ℓ − ; the hypergeometric solution gets multiplied by(1 − z ) − a/ .Pfaff’s fractional-linear transformation (2.3) gives a different relation between the upperparameters in (3.2) or (3.3); for example F a , a +12 + ℓ − k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z ! = (1 − z ) − a F a , − a − k − ℓ − k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) zz − ! . (3.11)The alternative shape of the upper parameters is convenient in Section 6. In the special cases k = 0 or ℓ = 0, one of the local exponents of dihedral hypergeometricequation (2.1) is equal to 1 /
2. Then the terminating double F or F sums in (3.1)–(3.3)become terminating single F sums. For example, special cases of (3.2)–(3.3) are F a , a +1212 − k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z ! =(1+ √ z ) − a F (cid:18) − k, a − k (cid:12)(cid:12)(cid:12)(cid:12) √ z √ z (cid:19) + (1 −√ z ) − a F (cid:18) − k, a − k (cid:12)(cid:12)(cid:12)(cid:12) √ z √ z − (cid:19) , (4.1) (cid:0) a +12 (cid:1) ℓ (cid:0) (cid:1) ℓ F a , a +12 + ℓ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z ! =(1+ √ z ) − a F (cid:18) − ℓ, a − ℓ (cid:12)(cid:12)(cid:12)(cid:12) √ z (cid:19) + (1 −√ z ) − a F (cid:18) − ℓ, a − ℓ (cid:12)(cid:12)(cid:12)(cid:12) − √ z (cid:19) , (4.2)( − k ( a ) k +1 k +1 (cid:0) (cid:1) k (cid:0) (cid:1) k +1 z k + F a +12 + k, a + k + 1 + k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z ! =(1 −√ z ) − a F (cid:18) − k, a − k (cid:12)(cid:12)(cid:12)(cid:12) √ z √ z − (cid:19) − (1+ √ z ) − a F (cid:18) − k, a − k (cid:12)(cid:12)(cid:12)(cid:12) √ z √ z (cid:19) , (4.3)2 (cid:0) a (cid:1) ℓ +1 (cid:0) (cid:1) ℓ √ z F a +12 , a + ℓ + 1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z ! =(1 −√ z ) − a F (cid:18) − ℓ, a − ℓ (cid:12)(cid:12)(cid:12)(cid:12) −√ z (cid:19) − (1+ √ z ) − a F (cid:18) − ℓ, a − ℓ (cid:12)(cid:12)(cid:12)(cid:12) √ z (cid:19) . (4.4)10his appearance of terminating F sums is expectable, since quadratic transformation(2.12) leads to a hypergeometric equation with a cyclic monodromy group. We havequadratic transformations between the dihedral and terminating F functions. In par-ticular, the identification z = 4 x/ (1 + x ) in classical formula (2.5) gives F (cid:18) a , a +12 a + k + 1 (cid:12)(cid:12)(cid:12)(cid:12) z (cid:19) = (cid:18) √ − z (cid:19) − a F (cid:18) − k, aa + k + 1 (cid:12)(cid:12)(cid:12)(cid:12) − √ − z √ − z (cid:19) , (4.5)while formulas (2.4) and (2.3) give F (cid:18) a , a +12 + ℓa + ℓ + 1 (cid:12)(cid:12)(cid:12)(cid:12) z (cid:19) = (cid:18) √ − z (cid:19) − a F (cid:18) − ℓ, aa + ℓ + 1 (cid:12)(cid:12)(cid:12)(cid:12) √ − z − √ − z + 1 (cid:19) . (4.6)We recognize here special cases of (3.1) after the substitution z − z and applicationof Pfaff’s fractional-linear transformation (2.3) to the terminating F series. Relatedly,formulas (3.8), (3.9) are standard hypergeometric identities when k = 0 or ℓ = 0.The dihedral functions with k = ℓ can be reduced to the considered case k ℓ = 0 viaa quadratic transformation. The quadratic transformation is E (cid:0) , k + , λ (cid:1) ←− E (cid:0) k + , k + , λ (cid:1) . (4.7)Quadratic transformations (2.5) and (2.6) give the following expressions: F a, a + k + − k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z ! = (1 + z ) − a F a , a +1212 − k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z (1 + z ) ! = (1+ √ z ) − a F (cid:18) − k, a − k (cid:12)(cid:12)(cid:12)(cid:12) √ z (1+ √ z ) (cid:19) + (1 −√ z ) − a F (cid:18) − k, a − k (cid:12)(cid:12)(cid:12)(cid:12) − √ z ( √ z − (cid:19) , (4.8) F (cid:18) a, a + k + a + 2 k + 1 (cid:12)(cid:12)(cid:12)(cid:12) − z (cid:19) = (cid:18) z (cid:19) − a F (cid:18) a , a +12 a + k + 1 (cid:12)(cid:12)(cid:12)(cid:12) (1 − z ) (1 + z ) (cid:19) = z k/ (cid:18) √ z (cid:19) − a − k F (cid:18) − k, k + 1 a + k + 1 (cid:12)(cid:12)(cid:12)(cid:12) − ( √ z − √ z (cid:19) . (4.9)Comparing with formulas (3.2), (3.1) for the left-hand sides here, we conclude F (cid:18) a ; − k, − k − k, − k (cid:12)(cid:12)(cid:12)(cid:12) x, − x (cid:19) = (cid:0) a + (cid:1) k (cid:0) (cid:1) k F (cid:18) − k, a − k (cid:12)(cid:12)(cid:12)(cid:12) x (2 − x ) (cid:19) , (4.10) F (cid:18) − k, − k ; k + 1 , k + 12 a + 2 k + 1 (cid:12)(cid:12)(cid:12)(cid:12) x, x x − (cid:19) = F (cid:18) − k, k + 1 a + k + 1 (cid:12)(cid:12)(cid:12)(cid:12) x x − (cid:19) . (4.11)11he finite F sums can be modified using these transformation formulas [7, Section 7]: F (cid:18) − k, a − k (cid:12)(cid:12)(cid:12)(cid:12) x (cid:19) = (1 − x ) k F (cid:18) − k, − a − k − k (cid:12)(cid:12)(cid:12)(cid:12) xx − (cid:19) (4.12)= k ! ( a ) k (2 k )! x k F (cid:18) − k, k + 11 − a − k (cid:12)(cid:12)(cid:12)(cid:12) x (cid:19) (4.13)= k ! (1 + a + k ) k (2 k )! x k F (cid:18) − k, k + 11 + a + k (cid:12)(cid:12)(cid:12)(cid:12) − x (cid:19) (4.14)= k ! (1 + a + k ) k (2 k )! F (cid:18) − k, a a + k (cid:12)(cid:12)(cid:12)(cid:12) − x (cid:19) (4.15)= k ! ( a ) k (2 k )! ( x − k F (cid:18) − k, − a − k − a − k (cid:12)(cid:12)(cid:12)(cid:12) − x (cid:19) . (4.16)These six hypergeometric expressions have the following arguments after the substitution x = 2 √ z (cid:14) (1 + √ z ), respectively:2 √ z √ z , √ z √ z − , √ z √ z , √ z − √ z , √ z − √ z + 1 , √ z + 1 √ z − . (4.17)It is easy to substitute further z /z or z − z here. The substitution z z/ ( z − z − p z − z, z + 2 p z − z,
12 + √ z − z z , − √ z − z z , − z + 2 p z − z, − z − p z − z. (4.18)The argument on the right-hand side of (4.6) can be written as √ − z − √ − z + 1 = 1 − z + 2 √ − zz . (4.19) Remark 4.1.
As well known [2, Section 2.9], there are generally 24 hypergeometricseries that are solutions of the same hypergeometric equation (2.1); they are referredto as 24
Kummer’s solutions . Particularly, a general hypergeometric equation has abasis of hypergeometric solutions at each of the three singular points; the six solutionsare different functions, and each of them has 4 representations as Gauss hypergeometricseries due to Euler-Pfaff transformations (2.2)–(2.3). When terminating or logarithmicsolutions are present, this structure of 24 solutions degenerates [7].The hypergeometric equation on the left-hand side of transformation (2.12) has adegenerate structure of 24 Kummer’s solutions, as exemplified by formulas (4.12)–(4.16).According to [7, Section 7], the same hypergeometric equation has other terminatingsolution(1 − x ) − a − k F (cid:18) − k, − a − k − k (cid:12)(cid:12)(cid:12)(cid:12) x (cid:19) = (1 − x ) − a F (cid:18) − k, a − k (cid:12)(cid:12)(cid:12)(cid:12) xx − (cid:19) . (4.20)12hen written in terms of z under the identification x = 2 √ z (cid:14) (1 + √ z ), both terminat-ing solutions are related (up to a power factor) by the conjugation √ z
7→ −√ z . Thetwo different terminating solutions are present on the left-hand side of (4.1). Both ter-minating solutions are representable by 6 terminating and 4 non-terminating F sums.The remaining 4 Kummer’s solutions of the transformed equation are non-terminating F series at x = 0. They are related by Euler-Pfaff transformations (2.2)–(2.3), andrepresent the following solution:( − k k ! ( a ) k +1 (2 k )! (2 k + 1)! x k +12 F (cid:18) k + 1 , a + 2 k + 12 k + 2 (cid:12)(cid:12)(cid:12)(cid:12) x (cid:19) =(1 − x ) − a − k F (cid:18) − k, − a − k − k (cid:12)(cid:12)(cid:12)(cid:12) x (cid:19) − F (cid:18) − k, a − k (cid:12)(cid:12)(cid:12)(cid:12) x (cid:19) . (4.21)Quadratic transformation (2.6) gives the following identification: F (cid:18) k + 1 , a + 2 k + 12 k + 2 (cid:12)(cid:12)(cid:12)(cid:12) √ z √ z (cid:19) = (cid:0) √ z (cid:1) a F a +12 + k, a + k + ℓ + 1 + k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z ! , (4.22)consistent with (3.3). Formulas (2.2)–(2.6) for fractional-linear and quadratic transfor-mations may not hold when a degenerate set of 24 Kummer’s solutions is involved. Inparticular, the terminating F sums in (4.12) and (4.20) are not related by Euler-Pfafftransformations (2.2)–(2.3). As noted in [7, Lemma 3.1], the generic transformations arecorrect here only if exactly one of the F sums is interpreted as non-terminating (follow-ing Remark 2.3). The right-hand side of (4.21) can be seen as the difference between theterminating and non-terminating interpretations of F (cid:18) − k, a − k (cid:12)(cid:12)(cid:12)(cid:12) x (cid:19) . Relatedly, there areno two-term identities between the dihedral function in (4.5) and the terminating F sums in (4.12) or (4.20).Transformations (2.12) and (4.7) involve degenerate Gauss hypergeometric functionson both sides when λ is an integer. We comment further on degeneracies of Kummer’s24 solutions in Subsection sc:degenerate below. Remark 4.2.
The dihedral functions with k = 0 or ℓ = 0 can be expressed in terms ofthe (associated) Legendre functions P µν ( z ), Q µν ( z ) with integer ν or half-integer µ . Therelation to the Legendre functions with integer ν is clear after comparing the terminatingseries in (4.13) with the definition [1, 8.1.2]: P µk ( x ) = 1Γ(1 − µ ) (cid:18) x + 1 x − (cid:19) µ F (cid:18) − ν, ν + 11 − µ (cid:12)(cid:12)(cid:12)(cid:12) − x (cid:19) . (4.23)13y formulas in [1, Chapter 8] or [4, Chapter III] we obtain these expressions:(2 k − − a ) z − k (1 − z ) a + k F a , a +1212 − k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z ! = 12 P a + kk (cid:18) √ z (cid:19) + ( − k P a + kk (cid:18) − √ z (cid:19) (4.24)= P a + kk (cid:18) √ z (cid:19) − sin πa ( − a π Q a + kk (cid:18) √ z (cid:19) , (4.25) z k +12 (1 − z ) a + k F a +12 , a + k + 1 + k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z ! = (2 k + 1)!!Γ( a + 2 k + 1) ( − − a − k Q a + kk (cid:18) √ z (cid:19) , (4.26)2 ℓ +1 (cid:0) a +12 (cid:1) ℓ (1 − z ) a + ℓ ( − a − ℓ Γ(1 − a ) F a , a +12 + ℓ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z ! = P a + ℓk (cid:0) √ z (cid:1) + ( − − a − ℓ P a + ℓk (cid:0) −√ z (cid:1) , (4.27)2 ℓ +2 (cid:0) a (cid:1) ℓ +1 (1 − z ) a + ℓ √ z ( − a − ℓ Γ(1 − a ) F a +12 , a + ℓ + 1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z ! = P a + ℓk (cid:0) √ z (cid:1) − ( − − a − ℓ P a + ℓk (cid:0) −√ z (cid:1) , (4.28) z − k (cid:18) − z (cid:19) a + k F (cid:18) a , a +12 a + k + 1 (cid:12)(cid:12)(cid:12)(cid:12) − z (cid:19) = Γ(1 + a + k ) P − a − kk (cid:18) √ z (cid:19) . (4.29)We generally mean ( − x = exp( iπx ) here. For integer k , formulas [1, 8.2.3, 8.2.5] givethe simpler relation P − µk ( x ) = ( − k Γ(1 − µ + k )Γ(1 + µ + k ) P µk ( − x ) , (4.30)while [1, 8.2.4, 8.2.6] give Q µk ( − x ) = ( − k +1 Q µk ( x ) , Q − µk ( x ) = ( − − µ Γ(1 − µ + k )Γ(1 + µ + k ) Q µk ( x ) . (4.31)Expressions in terms of Legendre functions with half-integer µ are obtained by applying[1, 8.2.7–8]: P − k − − µ − (cid:18) √ − z (cid:19) = ( − − µ p /π Γ( µ + k + 1) (cid:18) − zz (cid:19) Q µk (cid:18) √ z (cid:19) ,Q − k − − µ − (cid:18) √ − z (cid:19) = i ( − k +1 Γ( − µ − k ) r π (cid:18) − zz (cid:19) P µk (cid:18) √ z (cid:19) . So far we considered solutions of hypergeometric equations E ( k + 1 / , ℓ + 1 / , λ ), where k, ℓ are integers but λ is not an integer. If the third local exponent difference λ is aninteger, the monodromy group is either completely reducible and isomorphic to Z / Z or(in presence of logarithmic solutions) it is isomorphic to an infinite dihedral group. Firstwe state conditions how to separate the two cases.14 .1 Conditions for logarithmic solutions Theorem 5.1.
Let k, ℓ, n denote non-negative integers. Then equation E ( k + 1 / , ℓ + 1 / , n ) has logarithmic solutions if and only if n ≤ k + ℓ, if n + k + ℓ is even , (5.1) n < | k − ℓ | , if n + k + ℓ is odd . (5.2) If this is the case, the monodromy group of ( ) is an infinite dihedral group; otherwisethe monodromy group is isomorphic to Z / Z .Proof. A representative equation (2.1) with the assumed local exponents has A = − n + k + ℓ , B = − n + k − ℓ − , C = 12 − k. The sequence A , 1 − B , C − A , 1 + B − C contains exactly two integers. By part (3) of[7, Theorem 2.2], there are no logarithmic solutions precisely when the two integers areeither both positive or both non-positive. Equivalently, there are logarithmic solutionsprecisely when one of the integers is positive while the other is non-positive. If n + k + ℓ is even, the two integers are A = − n + k + ℓ B − C = 1 + k + ℓ − n . The first integer is always zero or negative; the second integer is positive exactly when n ≤ k + ℓ . If n + k + ℓ is odd, the two integers are1 − B = 1 + n + k − ℓ C − A = 1 + n − k + ℓ . We may assume ℓ ≤ k without loss of generality. Then the first integer is positive; thesecond integer is non-positive exactly when n < k − ℓ .For comparison, recall [7, Section 9] that hypergeometric equation E ( k, ℓ, n ) withnon-negative integers k, ℓ, n has logarithmic solutions if and only if one of the integersis greater than the sum of the other two. Here is a more direct formulation of Theorem5.1. Corollary 5.2.
Suppose that p, q are half-integers, and n is a non-negative integer. Theset {| p − q | , | p + q |} contains two integers of different parity; let K be the integer in thisset such that K + n is odd. Then equation E ( p, q, n ) has the monodromy group isomorphicto Z / Z if K < n , and it has logarithmic solutions otherwise.
Here is a formulation of Theorem 5.1 that refers to the parameter a = λ + k + ℓ in(2.10) rather than to the local exponent difference λ .15 orollary 5.3. Let k, ℓ, m denote non-negative integers, and suppose that k ≤ ℓ . Hyper-geometric equation ( ) with ( ) and a = − m has logarithmic solutions if and onlyif ≤ m ≤ k + ℓ, for even m, (5.3) k < m + 12 ≤ ℓ, for odd m. (5.4) If this is the case, the monodromy group of ( ) is an infinite dihedral group; otherwisethe monodromy group is isomorphic to Z / Z .Proof. Theorem 5.1 is being applied to equation E ( k + 1 / , ℓ + 1 / , m − k − ℓ ) or to E ( k + 1 / , ℓ + 1 / , k + ℓ − m ).We refer to solutions of a hypergeometric equation with the monodromy group Z / Z as degenerate . Degenerate or logarithmic solutions are involved exactly when a Pochham-mer factor on the left-hand side of our main identities (3.2)–(3.3) vanishes. The identitiesare still valid even if the left-hand side vanishes. Recalling λ = a − k − ℓ , we set K = min( k, ℓ ) , K = ℓ − k, L = k + ℓ. (5.5)Then Pochhammer factors in both (3.2) and (3.3) vanish if and only if a ∈ {− , − , . . . , − K } , (5.6) λ ∈ {| K | + 1 , | K | + 3 , . . . , L − } . (5.7)Only a Pochhammer factor in (3.3) vanishes if and only if a ∈ { , − , . . . , − L ) } ∪ {− − K , − − K , . . . , − k } , (5.8) λ ∈ {− L, − L, . . . , L } ∪ { K + 1 , K + 3 , . . . , | K | − } , (5.9)and only the Pochhammer factor in (3.2) vanishes if and only if a ∈ {− − K , − − K , . . . , − ℓ } , (5.10) λ ∈ {− K + 1 , − K + 3 , . . . , | K | − } . (5.11)By Theorem 5.1, hypergeometric equation E ( k +1 / , ℓ + 1 / , λ ) has logarithmic solutionsif and only if exactly one of conditions (5.9) or (5.11) holds. Equivalently, by Corollary5.3 we have to check whether exactly one of conditions (5.8) or (5.10) holds. Z / Z By Corollary 5.3, the monodromy group of hypergeometric equation (2.1) with (2.10) and a = − m is isomorphic to Z / Z if and only if either (5.6) holds or both (5.8) and (5.10)16old. In the former case, formulas (3.2) and (3.3) do not give terminating expressionsfor the functions F − m , ℓ − m − − k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z ! , F k − m − , k + ℓ − m + 1 + k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z ! . (5.12)But terminating expressions for these functions are obtained by applying Euler’s trans-formation (2.2). Adding up the right-hand sides of (3.2) and (3.3) suggests the followinglemma. Lemma 5.4.
Suppose that m is an odd positive integer. Then for any b, c ∈ C theterminating sum F (cid:18) − m ; b, c b, c (cid:12)(cid:12)(cid:12)(cid:12) x, − x (cid:19) is identically zero.Proof. Here we have a triangular p ≥ q ≥ p + q ≤ m terminating sum (2.13), asopposed to rectangular 0 ≤ p ≤ k , 0 ≤ q ≤ ℓ terminating sums frequent in this article.The F ( x, − x ) sum is a rational function in b, c of degree at most 2 m . It is enoughto show that this rational function is zero for infinitely many values of b, c . By (3.2) and(3.3), the F sum is zero when b, c are integers ≤ − m +12 .We have the monodromy group Z / Z also in the cases a = − m with odd m > k, ℓ ) or even m > k + ℓ ). Then the hypergeometric functions in (5.12) themselvesterminate, while a non-terminating F solution is (3.4). Theorem 5.4 does not apply tothe F functions in (3.2)–(3.3) for odd m > k, ℓ ), because generallylim b →− k ( − m ) i + j ( b ) i ( c ) j i ! j ! (2 b ) i (2 c ) j = 0 for 2 k < i ≤ m (5.13)in the triangular sum of Theorem 5.4, while the same term is taken for zero in therectangular sums in (3.2)–(3.3). By Corollary 5.3, hypergeometric equation (2.1) with (2.10) and a = − m has logarithmicsolutions if and only if exactly one of the conditions (5.8) or (5.10) holds. Then one ofthe equations (3.2), (3.3) vanishes, and the other is a terminating hypergeometric sum.The terminating sum is a non-logarithmic solution, obviously. We have (cid:0) − m (cid:1) ℓ (cid:0) (cid:1) ℓ F − m , ℓ − m − − k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z ! = (1 − √ z ) m F (cid:18) − m ; − k, − ℓ − k, − ℓ (cid:12)(cid:12)(cid:12)(cid:12) √ z √ z − , √ z − (cid:19) (5.14)in the case (5.8) with a = − m , since both summands on the right-hand side of (3.2) areequal. We have the same right-hand side expression for the left-hand side of (3.3) in thecase (5.10) with a = − m . The F function in the vanishing equation (3.2) or (3.3) has alogarithmic expression that can be obtained by differentiating both sides of the vanishingequation with respect to a and evaluating at a = − m . We have dda (1 + √ z ) − a = − (1 + √ z ) − a log(1 + √ z )17nd similarly for dda (1 − √ z ) − a . On the left-hand side of the vanishing equation (3.2) or(3.3) we have exactly one (Pochhammer) factor vanishing; to differentiate the product,it is enough to differentiate only the vanishing Pochhammer symbol. Lemma 5.5.
The formula for differentiating a Pochhammer symbol is dda ( a ) N = ( ( a ) N (cid:16) a + a +1 + . . . + a + N − (cid:17) , if ( a ) N = 0 , ( − a ( − a )!( N + a − , if ( a ) N = 0 . (5.15) Proof.
If ( a ) N = 0, we are differentiating a product of N linear functions in a . If ( a ) N = 0then a is zero or a negative integer. Writing a = − m + ǫ we have( a ) N = ( − m + ǫ )(1 − m + ǫ ) · · · ( − ǫ ) ǫ (1 + ǫ ) · · · ( N − m − ǫ ) . (5.16)The differentiation is equivalent to dividing out ǫ and setting ǫ = 0.Let ψ ( x ) denote the digamma function, ψ ( x ) = Γ ′ ( x ) / Γ( x ). The sum a + a +1 + . . . + a + N − can be written as ψ ( a + N ) − ψ ( a ) if a is not zero or a negative integer. For aninteger m ≥ − m ) † N = dda ( a ) N (cid:12)(cid:12)(cid:12)(cid:12) a = − m . (5.17)By Lemma 5.5 we have( − m ) † N = (cid:26) ( − m ) N (cid:0) ψ ( m + 1 − N ) − ψ ( m + 1) (cid:1) if N ≤ m, ( − m m ! ( N − m − N > m. (5.18)Note that ( − m ) † N = 0 when N >
0. On the other hand, ( − m ) † = 0 even if m = 0.When differentiating both sides of (3.2) or (3.3) with respect to a , we keep in mindthat dda (cid:0) a (cid:1) N evaluates to (cid:0) − m (cid:1) † N . For even a = − m satisfying (5.8), we differentiate(3.3), use (5.14) and obtain (cid:0) − m (cid:1) k (cid:0) m (cid:1) ! (cid:0) k + ℓ − m (cid:1) ! (cid:0) (cid:1) k (cid:0) (cid:1) k +1 (cid:0) (cid:1) ℓ ( − k + m z k + F k − m − , k + ℓ − m + 1 + k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z ! = (cid:0) − m (cid:1) ℓ (cid:0) (cid:1) ℓ log 1 + √ z − √ z F − m , ℓ − m − − k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z ! +(1 − √ z ) m F † (cid:18) − m ; − k, − ℓ − k, − ℓ (cid:12)(cid:12)(cid:12)(cid:12) √ z √ z − , − √ z (cid:19) − (1 + √ z ) m F † (cid:18) − m ; − k, − ℓ − k, − ℓ (cid:12)(cid:12)(cid:12)(cid:12) √ z √ z ,
21 + √ z (cid:19) . (5.19)Here the F † functions are rectangular sums of ( k + 1)( ℓ + 1) terms, defined as the F sums in (2.13) but with each Pochhammer symbol ( − m ) i + j replaced by the derivative18 − m ) † i + j following (5.18). For odd a = − m satisfying (5.8), the differentiation of (3.3)gives the same right-hand side, but the left-hand side is (cid:0) m − (cid:1) ! (cid:0) k − m +12 (cid:1) ! (cid:0) − m (cid:1) k + ℓ +1 (cid:0) (cid:1) k (cid:0) (cid:1) k +1 (cid:0) (cid:1) ℓ ( − k + m − z k + F k − m − , k + ℓ − m + 1 + k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z ! . (5.20)For odd a = − m satisfying (5.10), differentiation of (3.2) gives (cid:0) m − (cid:1) ! (cid:0) ℓ − m +12 (cid:1) ! (cid:0) (cid:1) ℓ ( − m − F − m , ℓ − m − − k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z ! = (cid:0) − m (cid:1) k (cid:0) − m (cid:1) k + ℓ +1 (cid:0) (cid:1) k (cid:0) (cid:1) k +1 (cid:0) (cid:1) ℓ ( − k z k + log 1 + √ z − √ z F k − m − , k + ℓ − m + 1 + k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z ! +(1 + √ z ) m F † (cid:18) − m ; − k, − ℓ − k, − ℓ (cid:12)(cid:12)(cid:12)(cid:12) √ z √ z ,
21 + √ z (cid:19) +(1 − √ z ) m F † (cid:18) − m ; − k, − ℓ − k, − ℓ (cid:12)(cid:12)(cid:12)(cid:12) √ z √ z − , − √ z (cid:19) . (5.21)As we will see in Section 6, nice expressions for our dihedral F functions are obtainedunder the trigonometric substitution z
7→ − tan x . In particular, one may recognizearctan x = 12 i log 1 + ix − ix . (5.22) Kummer’s solutions
As observed in Remark 4.1, the structure of 24 Kummer’s solutions degenerates if thehypergeometric equation has terminating or logarithmic solutions. Particurlarly, not all24 Kummer’s solutions are well defined or distinct when logarithmic solutions are present.In the case of the monodromy group Z / Z , the degenerate structure is described in[7, Section 7]. The two solutions in (5.12) can be represented by terminating F sums in6 ways (with any of the arguments z, z/ ( z − , /z, / (1 − z ) , − /z, − z ) and by non-terminating F series (around z = 0 or z = ∞ ) in 4 ways. The remaining four of the 24Kummer’s solutions represent, up to a power factor, the function in (3.1). Terminating F expression (3.1) holds, but that function can be expressed as a linear combination oftwo terminating F solutions. For example, the following z − z version of [7, (43)]can be used:(1 − z ) n + m +12 F (cid:18) a + n + 1 , m + 1 n + m + 2 (cid:12)(cid:12)(cid:12)(cid:12) − z (cid:19) = ( n + 1) m +1 ( − a ) m +1 2 F (cid:18) − n, a − m a (cid:12)(cid:12)(cid:12)(cid:12) z (cid:19) + ( m + 1) n +1 ( a ) n +1 z − a F (cid:18) − m, − a − n − a (cid:12)(cid:12)(cid:12)(cid:12) z (cid:19) , (5.23)19ith ( n, m, a ) (cid:0) k − m +12 , ℓ − m +12 , − k − (cid:1) , for odd m < k, ℓ ) , ( n, m, a ) (cid:0) m − − ℓ, m − − k, − k − (cid:1) , for odd m > k, ℓ ) , ( n, m, a ) (cid:0) m , m − k − ℓ − , − k − (cid:1) , for even m > k + ℓ ) . The logarithmic case with terminating solutions is described in [7, Section 6]. Thereare 20 distinct hypergeometric series, or less if m = k + ℓ . Among them, there are 8terminating and 4 nonterminating hypergeometric series representing the non-logarithmicterminating solution, as in (5.14). The remaining 8 (in general) Kummer’s solutions arenon-terminating series around z = 0 or z = ∞ and represent two different functions, inparticular the logarithmic solution among (5.19)–(5.21). The logarithmic solution can beexpressed following [7, Theorem 6.1] as well. Formulas [7, (36) and (38)] give the genericidentity( − m +1 z n + m +1 − a (1 − a ) n + m +1 2 F (cid:18) m + 1 − a, n + m + 1 n + m + 2 − a (cid:12)(cid:12)(cid:12)(cid:12) z (cid:19) = ( m + 1 − a ) n n ! ( n + m )! F (cid:18) − n, aa − n − m (cid:12)(cid:12)(cid:12)(cid:12) z (cid:19) (log(1 − z ) + π cot πa ) − ( z − − m (1 − a ) m m − X j =0 ( a − m ) j ( m − j − n + m − j )! j ! (1 − z ) j + n X j =0 ( a ) j ( ψ ( a + j ) + ψ ( n − j +1) − ψ ( m + j +1) − ψ ( j +1))( m + j )! ( n − j )! j ! ( z − j +( − n ∞ X j = n +1 ( a ) j ( j − n − m + j )! j ! (1 − z ) j , (5.24)where we should substitute( n, m, a ) (cid:0) m , k + ℓ − m, ℓ − m − (cid:1) , if m is even, 0 ≤ m ≤ k + ℓ, ( n, m, a ) (cid:0) k + ℓ − m , m − k − ℓ, m +12 − k (cid:1) , if m is even, k + ℓ ≤ m ≤ k + ℓ ) , ( n, m, a ) (cid:0) m − − ℓ, k + ℓ − m, − m (cid:1) , if m is odd, 2 min( k, ℓ ) < m < k, ( n, m, a ) (cid:0) m − − k, k + ℓ − m, k + ℓ − m + 1 (cid:1) , if m is odd, 2 min( k, ℓ ) < m < ℓ, to get the same F function on the left-hand side as in (5.19)–(5.21). The cot πa termis zero when a is a half-integer in (5.24). These formulas do not contain double sums orsquare roots, but the last term in (5.24) is a non-terminating series. Remark 5.6.
As observed in Section 4, the terminating F or F sums in (3.1)–(3.3)become terminating F sums in the case k = 0 or ℓ = 0. If, additionally, a is an inte-ger, the structure of 24 Kummer’s solutions for the quadratically transformed equationdegenerates further. If a = − m with 0 ≤ m ≤ k , there are logarithmic solutions byCorollary 5.3. We have to apply then [7, Section 9] with ( n, m, ℓ ) = ( | k − m | , | k − m | , m )20o the transformed F sums. There are at most 10 different terminating F solutions,all representing the same elementary solution of the transformed equation.If a is an integer and | a + k | > k , the dihedral group degenerates to Z / Z , whilethe monodromy group of the quadratically transformed equation is trivial. By [7, Sec-tion 8], the transformed equation has 3 solutions like in (4.12), each representable by 6terminating and 2 non-terminating F series. Formulas (1.6)–(1.7) show attractive trigonometric expressions for the simplest dihedralfunctions. Analogous trigonometric expressions are possible for all dihedral functions,if only due to contiguous relations. The literature appears to give only trigonometricexpressions for the simplest dihedral functions. In particular [1, 15.1], [4, 2.8], here arefractional-linear transformations of (1.6)–(1.7): F a , − a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin x ! = cos ax cos x , (6.1) F a , − a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin x ! = 2 sin axa sin 2 x , (6.2) F a , a +1212 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − tan x ! = cos ax cos a x, (6.3) F a +12 , a +2232 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − tan x ! = sin axa sin x cos a +1 x. (6.4)According to the latter two formulas, we should use the argument substitution z tan x in our main expressions. Nicer expressions are obtained after fractional-lineartransformation (2.3), hence with the argument sin x . The following theorem gives trigonometric versions of formulas (3.2)–(3.3) after Pfaff’stransformation (2.3). Subsequently, we discuss here simplification of trigonometric for-mulas. Trigonometric modification of formula (3.1) is discussed in Subsection 6.4.
Theorem 6.1.
Let us denote Υ a,k,ℓp,q ( x ) := 2 p + q ( − k ) p ( − ℓ ) q ( a ) p + q ( − k ) p ( − ℓ ) q p ! q ! sin p x cos q x. (6.5)21 hen (cid:0) a +12 (cid:1) ℓ (cid:0) (cid:1) ℓ F a , − a − k − ℓ − k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin x ! = k X p =0 ℓ X q =0 Υ a,k,ℓp,q ( x ) cos (cid:0) ax + ( p + q ) x − π p (cid:1) , (6.6) (cid:0) a +12 (cid:1) k (cid:0) a (cid:1) k + ℓ +1 sin k +1 x (cid:0) (cid:1) k (cid:0) (cid:1) k +1 (cid:0) (cid:1) ℓ F a +12 + k, − a − ℓ + k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin x ! = k X p =0 ℓ X q =0 Υ a,k,ℓp,q ( x ) sin (cid:0) ax + ( p + q ) x − π p (cid:1) . (6.7) Proof.
After fractional-linear transformation (2.3) and the substitution z
7→ − tan x in(3.2) we get (cid:0) a +12 (cid:1) ℓ (cid:0) (cid:1) ℓ F a , − a − k − ℓ − k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin x ! = exp( − iax )2 F (cid:18) a ; − k, − ℓ − k, − ℓ (cid:12)(cid:12)(cid:12)(cid:12) i sin x exp ix , x exp ix (cid:19) + exp iax F (cid:18) a ; − k, − ℓ − k, − ℓ (cid:12)(cid:12)(cid:12)(cid:12) − i sin x exp( − ix ) , x exp( − ix ) (cid:19) . We use Euler’s formula exp ix = cos x + i sin x as well. We add the two double sums termby term, and after the identification ± i = 2 exp( ± i π ) we get the double sum in (6.6).Note that Υ a,k,ℓp,q ( x ) is the generic summand of F (cid:18) a ; − k, − ℓ − k, − ℓ (cid:12)(cid:12)(cid:12)(cid:12) x, x (cid:19) .Formula (6.7) follows similarly. Withal, the substitution z
7→ − tan x into ( − k z k +1 / gives i sin k +1 x/ cos k +1 x .The right-hand side of (6.6) can be written as P k,ℓ ( x ) cos ax + Q k,ℓ ( x ) sin ax, (6.8)with P k,ℓ ( x ) = k X p =0 ℓ X q =0 Υ a,k,ℓp,q ( x ) cos (cid:0) π p − ( p + q ) x (cid:1) ,Q k,ℓ ( x ) = k X p =0 ℓ X q =0 Υ a,k,ℓp,q ( x ) sin (cid:0) π p − ( p + q ) x (cid:1) . The right-hand side of (6.7) can be then written as P k,ℓ ( x ) sin ax − Q k,ℓ ( x ) cos ax, (6.9)and we have P ℓ,k ( x ) = P k,ℓ ( π − x ) , Q ℓ,k ( x ) = − Q k,ℓ ( π − x ) . (6.10)22enerally, here are the first few terms of the double sums P k,ℓ ( x ), Q k,ℓ ( x ): P k,ℓ ( x ) = 1 + a cos x + a ( a + 1)( ℓ − ℓ − x cos 2 x + a ( a + 1)( a + 2)( ℓ − ℓ −
1) cos x cos 3 x + . . . + a sin x + a ( a + 1)2 sin x + a ( a + 1)( a + 2)( ℓ − ℓ − x cos x sin 3 x + . . . − a ( a + 1)( k − k − x cos 2 x − a ( a + 1)( a + 2)( k − k − x cos x cos 3 x + . . . − a ( a + 1)( a + 2)( k − k −
1) sin x sin 3 x − . . . + . . . ,Q k,ℓ ( x ) = 0 − a x − a ( a + 1)( ℓ − ℓ − x sin 2 x − a ( a + 1)( a + 2)( ℓ − ℓ −
1) cos x sin 3 x − . . . + a x + a ( a + 1)4 sin 4 x + a ( a + 1)( a + 2)( ℓ − ℓ − x cos x cos 3 x + . . . + a ( a + 1)( k − k − x sin 2 x + a ( a + 1)( a + 2)( k − k − x cos x sin 3 x + . . . − a ( a + 1)( a + 2)( k − k −
1) sin x cos 3 x − . . . − . . . . Evidently, significant simplification in the double sums P k,ℓ ( x ), Q k,ℓ ( x ) is possible if wecollect terms to the same Pochhammer products ( a ) p + q . If both k ≥ ℓ ≥
3, wehave P k,ℓ ( x ) = 1 + a + a ( a + 1)2 (cid:0) − B k,ℓ ( x ) cos 2 x (cid:1) + a ( a + 1)( a + 2)6 (cid:0) − B k,ℓ ( x ) cos 2 x (cid:1) + . . . ,Q k,ℓ ( x ) = a ( a + 1)2 B k,ℓ ( x ) sin 2 x + a ( a + 1)( a + 2)2 B k,ℓ ( x ) sin 2 x + . . . , where B k,ℓ ( x ) = cos x ℓ − − sin x k − . Simplification in the symmetric case k = ℓ is particularly significant, as could beexpected from relation (4.8) to the case ℓ = 0. Quadratic transformation (2.4) can bewritten as F (cid:18) a, b a + b +12 (cid:12)(cid:12)(cid:12)(cid:12) sin x (cid:19) = F a , b a + b +12 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin x ! . (6.11)This relates the case ℓ = k with a a of (6.6) to the case ℓ = 0 with x → x of (6.6).To use (6.11) similarly with (6.7), one has to apply Euler’s transformation (2.2) to the F function in (6.7) first.For the dihedral functions with small k , ℓ nice expressions are obtained if we makethe substitution a λ − k − ℓ and write the right-hand sides of (6.6) and (6.7) as,23espectively, P k,ℓ ( x ) cos λx + Q k,ℓ ( x ) sin λx, (6.12) P k,ℓ ( x ) sin λx − Q k,ℓ ( x ) cos λx. (6.13)Explicitly, we have the rotation transformation P k,ℓ ( x ) = P k,ℓ ( x ) cos( k + ℓ ) x − Q k,ℓ sin( k + ℓ ) x, Q k,ℓ ( x ) = P k,ℓ ( x ) sin( k + ℓ ) x + Q k,ℓ cos( k + ℓ ) x. (6.14)A bit tedious translation of (6.10) shows the following effect of interchange of the indices k, ℓ : P ℓ,k ( x ) = (cid:26) ( − j P k,ℓ ( π − x ) , if k + ℓ = 2 j even , ( − j Q k,ℓ ( π − x ) , if k + ℓ = 2 j + 1 odd , (6.15) Q ℓ,k ( x ) = (cid:26) ( − j − Q k,ℓ ( π − x ) , if k + ℓ = 2 j even , ( − j P k,ℓ ( π − x ) , if k + ℓ = 2 j + 1 odd . (6.16)But here are attractive evaluation formulas after the substitution a λ − k − ℓ : F − λ , − − λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin x ! = cos x cos λx + sin x sin λxλ , (6.17) F − λ , − − λ − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin x ! = cos x cos λx + λ sin x sin λx, (6.18) F λ , − λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin x ! = λ cos x sin λx − sin x cos λx ( λ −
1) sin x , (6.19) F λ , − λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin x ! = 3 cos x sin λx − λ sin x cos λxλ ( λ −
1) ( λ + 1) sin x , (6.20) F − λ , − − λ − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin x ! = cos 2 x cos λx + λ x sin λx, (6.21) F λ , − λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin x ! = 3 cos 2 x sin λx − λ sin 2 x cos λxλ ( λ −
2) ( λ + 2) sin x . (6.22)24ome further expressions: P , ( x ) = λ cos x − x − , Q , ( x ) = λ sin 2 x , P , ( x ) = λ ( λ − x (2 cos 2 x − , Q , ( x ) = λ −
23 sin x ( λ cos x − x ) , P , ( x ) = ( λ − λ − (cid:18) cos 4 x + 2 − λ sin x (cid:19) , Q , ( x ) = λ ( λ −
1) ( λ − x, P , ( x ) = λ cos x (cid:0) λ cos x −
11 sin x − (cid:1) , Q , ( x ) = sin x (2 λ sin x − x − , P , ( x ) = λ − (cid:0) λ cos x (1 − x ) + cos 4 x + 16 sin x − (cid:1) , Q , ( x ) = λ ( λ − x ( λ cos x −
14 sin x − , P , ( x ) = ( λ − λ − λ − x (cid:18) cos 4 x + 4 − λ sin x (cid:19) , Q , ( x ) = λ ( λ −
1) ( λ −
3) ( λ − x (cid:18)
11 cos 4 x + 19 − λ sin x (cid:19) , P , ( x ) = λ cos x − λ cos x (7 sin x + 2)21 + cos 4 x + 32 sin x + 235 , Q , ( x ) = λ sin 2 x (2 λ cos x −
10 sin x − . Because of the O ( x k +1 ) solution in (6.7), the introduced trigonometric polynomials haveproperties resembling simultaneous Pad´e approximation. We have Q k,ℓ ( x ) P k,ℓ ( x ) = tan ax + O (cid:0) x k +1 (cid:1) , Q k,ℓ ( x ) P k,ℓ ( x ) = tan λx + O (cid:0) x k +1 (cid:1) . (6.23)and by the symmetry (6.10) or (6.15)–(6.16) of k, ℓ , Q k,ℓ ( π − x ) P k,ℓ ( π − x ) = − tan ax + O (cid:0) x ℓ +1 (cid:1) , (cid:18) P k,ℓ ( π − x ) Q k,ℓ ( π − x ) (cid:19) ε = ε tan λx + O (cid:0) x ℓ +1 (cid:1) . (6.24)Here ε = ( − k + ℓ − . It is however not clear what characterizes the trigonometric polyno-mials uniquely in the approximation context. From the degenerate cases with vanishing25ochhammer factors on the right-hand sides of (6.6) or (6.7) we get the following van-ishing or exact approximation properties: P k,ℓ ( x ) = 0 , Q k,ℓ ( x ) = 0 , if (5.6) holds ,Q k,ℓ ( x ) P k,ℓ ( x ) = tan ax, if (5.8) holds ,P k,ℓ ( x ) Q k,ℓ ( x ) = − tan ax, if (5.10) holds , (6.25) P k,ℓ ( x ) = 0 , Q k,ℓ ( x ) = 0 , if (5.7) holds , Q k,ℓ ( x ) P k,ℓ ( x ) = tan λx, if (5.9) holds , P k,ℓ ( x ) Q k,ℓ ( x ) = − tan λx, if (5.11) holds , and similarly with P k,ℓ ( π − x ) , Q k,ℓ ( π − x ) or P k,ℓ ( π − x ) , Q k,ℓ ( π − x ). These conditionsdo not determine the trigonometric polynomials uniquely either, because P k,ℓ ( x ) , Q k,ℓ ( x )are not the shifted a λ − k − ℓ versions of P k,ℓ ( x ) , Q k,ℓ ( x ). Contiguous relations for the considered F functions translate into the following recur-rence relations: P k,ℓ ( x ) = a + 2 k − k − x P k − ,ℓ ( x ) + a + 2 ℓ − ℓ − x P k,ℓ − ( x ) , (6.26) (cid:16) a k + ℓ + 1 (cid:17) P k,ℓ ( x ) = (cid:18) k + 12 (cid:19) P k +1 ,ℓ ( x ) + (cid:18) ℓ + 12 (cid:19) P k,ℓ +1 ( x ) , (6.27) (cid:18) ℓ + 12 (cid:19) P k,ℓ +1 ( x ) = (cid:18) ℓ + 12 + ( a + k + ℓ ) cos x (cid:19) P k,ℓ ( x ) − a + 2 ℓ − ℓ − (cid:16) a k + ℓ (cid:17) cos x P k,ℓ − ( x ) . (6.28)The recurrence relations for Q k,ℓ ’s are the same. Three-term recurrences for the trigono-metric polynomials P k,ℓ ( x ), Q k,ℓ ( x ) are less clear, because some shifts in k, ℓ lead tohalf-integer shifts in the upper parameters of the F functions, and transformation (6.14)mixes P k,ℓ ’s with Q k,ℓ ’s. Since we formally have Q , ( x ) = 0, Q , − ( x ) = 0, three-termrecurrences between these polynomials with minimal shifts in k, ℓ should not be expected.26ere are some consequences of the contiguous relations, nevertheless: (cid:0) k cos x + 2 ℓ sin x + 1 (cid:1) P k,ℓ ( x ) = 2 k + 12 ℓ − λ − k + ℓ −
1) cos x P k +1 ,ℓ − ( x )+ 2 ℓ + 12 k − λ + k − ℓ −
1) sin x P k − ,ℓ +1 ( x ) , (6.29)sin x k + 1 P k,ℓ ( x ) + P k +1 ,ℓ +1 ( x ) λ − k − ℓ − λ − k + ℓ − ℓ − ℓ + 1) cos x P k +1 ,ℓ − ( x ) , (6.30)(2 ℓ − P k,ℓ ( x ) = λ − ( k + ℓ ) (2 k − k + 1) ( λ − k − ℓ + 1) sin x P k − ,ℓ − ( x )+( λ − k + ℓ − P k +1 ,ℓ − ( x ) . (6.31) ( ) A trigonometric version of formula (3.1) is pointless, because the substitution z
7→ − tan x does not give a meaningful F argument for a local expansion. We may however use thehyperbolic tangent substitution z tanh x (plus a λ − k − ℓ for a little neatness),and consider the approximation x → + ∞ : F (cid:18) λ − k − ℓ , λ − k + ℓ +12 λ + 1 (cid:12)(cid:12)(cid:12)(cid:12) x (cid:19) = 2 λ cosh λ x (cosh λx − sinh λx ) tanh k x × F (cid:18) k + 1 , ℓ + 1; − k, − ℓλ + 1 (cid:12)(cid:12)(cid:12)(cid:12) − coth x , − tanh x (cid:19) . (6.32)Trigonometric expressions for the dihedral functions with kℓ = 0 or k = ℓ could beobtained in relation to the Legendre functions (as described in Remark 4.2). Infinitetrigonometric series for general Legendre functions are given in [1, 8.7.1–8.7.2]. The use of trigonometric arguments is instructive for degenerate or logarithmic dihedral F functions as well, as suggested by formulas (1.4), (1.5). In the logarithmic cases (5.8),(5.10) the non-logarithmic solution like (5.14) in (3.2) or (3.3) is, respectively, P k,ℓ ( x )cos ax , − Q k,ℓ ( x )cos ax , (6.33)due to (6.25). The logarithmic solution is obtained by differentiating (6.9) or (6.8) withrespect to a . Under condition (5.8), the left hand side of (5.19) if a is even, or (5.20) if a is odd, is equal to x P k,ℓ ( x )cos ax + dP k,ℓ ( x ) da sin ax − dQ k,ℓ ( x ) da cos ax, with a = − m. (6.34)Under condition (5.10), the left hand side of (5.21) is equal to x Q k,ℓ ( x )cos ax + dP k,ℓ ( x ) da cos ax + dQ k,ℓ ( x ) da sin ax, with a = − m. (6.35)27imilar expressions can be obtained with the more compact trigonometric polynomials P k,ℓ ( x ) , Q k,ℓ ( x ), which depend on the parameter λ = a + k + ℓ . References [1] M. Abramowitz and I.A. Stegun.
Handbook of mathematical functions, with formu-las, graphs and mathematical tables . Number 55 in Applied Math. Series. Nat. Bur.Standards, Washington, D.C., 1964.[2] G.E. Andrews, R. Askey, and R. Roy.
Special Functions . Cambridge Univ. Press,Cambridge, 1999.[3] F. Beukers. Gauss’ hypergeometric function. In W. Abikoff et al, editor,
TheMathematical Legacy of Wilhelm Magnus: Groups, geometry and special functions ,volume 169 of
Contemporary Mathematics series , pages 29–43. AMS, Providence,2007.[4] A. Erd´elyi, editor.
Higher Transcendental Functions , volume I. McGraw-Hill BookCompany, New-York, 1953.[5] M. Petkovˇsek, H.S. Wilf, and D. Zeilberger.
A=B . A. K. Peters, Wellesley, Mas-sachusetts, 1996.[6] H. M. Srivastava and P. W. Karlsson.
Multiple Gaussian Hypergeometric Series .Ellis Horwood Ltd., 1985.[7] R. Vidunas. Degenerate Gauss hypergeometric functions.
Kyushu Journal of Math-ematics , 61:109–135, 2007. Available at http://arxiv.org/math.CA/0407265 .[8] R. Vid¯unas. Algebraic transformations of Gauss hypergeometricfunctions.
Funkcialaj Ekvacioj , 52(2):139–180, 2009. Available at http://arxiv.org/math.CA/0408269 .[9] R. Vidunas. Specialization of Appell’s functions to univariate hypergeometric func-tions.
Journal of Mathematical Analysis and Applications , 355:145–163, 2009. Avail-able at http://arxiv.org/abs/0804.0655 .[10] R. Vidunas. Transformations and invariants for dihedral Gauss hypergeometricfunctions. Available at http://arxiv.org/abs/1101.3688 ., 2011.[11] Wikipedia. Hypergeometric series. http://en.wikipedia.org/wiki/Hypergeometric serieshttp://en.wikipedia.org/wiki/Hypergeometric series