Comment on "Fourier transform of hydrogen-type atomic orbitals'', Can. J. Phys. Vol.\ 96, 724 - 726 (2018) by N. Yükçü and S. A. Yükçü
aa r X i v : . [ m a t h - ph ] A p r Comment on“Fourier transform of hydrogen-type atomic orbitals”,Can. J. Phys. Vol. 96, 724 - 726 (2018)by N. Y ¨ukc¸ ¨u and S. A. Y ¨ukc¸ ¨u
Ernst Joachim Weniger ∗ Institut f¨ur Physikalische und Theoretische Chemie, Universit¨at Regensburg, D-93040 Regensburg, Germany (Dated: In Press, Canadian Journal of Physics, April 30, 2019)Podolsky and Pauling (Phys. Rev. , 109 - 116 (1929)) were the first ones to derive an explicit expressionfor the Fourier transform of a bound-state hydrogen eigenfunction. Y¨ukc¸ ¨u and Y¨ukc¸ ¨u, who were apparentlyunaware of the work of Podolsky and Pauling or of the numerous other earlier references on this Fourier trans-form, proceeded differently. They expressed a generalized Laguerre polynomial as a finite sum of powers, orequivalently, they expressed a bound-state hydrogen eigenfunction as a finite sum of Slater-type functions. Thisapproach looks very simple, but it leads to comparatively complicated expressions that cannot match the sim-plicity of the classic result obtained by Podolsky and Pauling. It is, however, possible to reproduce not onlythe Podolsky and Pauling formula for the bound-state hydrogen eigenfunction, but to obtain results of similarquality also for the Fourier transforms of other, closely related functions such as Sturmians, Lambda functionsor Guseinov’s functions by expanding generalized Laguerre polynomials in terms of so-called reduced Besselfunctions. PACS numbers: 02.30.Gp, 03.65.Ge, 31.15.pKeywords: bound-state hydrogen eigenfunctions, Fourier transform, generalized Laguerre polynomials, hypergeometric se-ries, Slater-type functions, reduced Bessel functions
I. INTRODUCTION
Y¨ukc¸ ¨u and Y¨ukc¸ ¨u [2] derived explicit expressions for theFourier transform of a bound-state hydrogen eigenfunction.Their article creates the impression that their results [2, Eqs.(15) and (16)] are new. This is wrong. Moreover, their ex-plicit expressions are less compact and therefore also less use-ful than those already described in the literature.In 1929, Podolsky and Pauling [1, Eq. (28)] were the firstones to derive an explicit expression via a direct Fouriertransformation of a generating function of the generalizedLaguerre polynomials. In 1932, Hylleraas [3, Eqs. (11c)and (12)] derived this Fourier transformation algebraicallyby solving a differential equation for the momentum spaceeigenfunction. In 1935, Fock [4] re-formulated the momen-tum space Schr¨odinger equation for the hydrogen atom asa 4-dimensional integral equation, whose solutions – the 4-dimensional hyperspherical harmonics – are nothing but theFourier transforms of bound-state hydrogen eigenfunctions indisguise (see for example [5, Section VI] or the books by Av-ery [6, 7], Avery and Avery [8], and Avery, Rettrup, and Avery[9] and references therein).The Fourier transform of a bound state hydrogen eigenfunc-tion has been treated in numerous books and articles. Exam-ples are the books by Bethe and Salpeter [10, Eq. (8.8)], En-glefield [11, Eqs. (5.5) and (5.6)], or Biedenharn and Louck[12, Eq. (7.4.69)] or the relatively recent review by Hill [13,Eq. (9.55)]. This Fourier transform was even discussed in aWikipedia article [14], which cites the book by Bransden andJoachain [15, Eq. (A5.34)] as its source. There are also articles ∗ [email protected], [email protected] by Klein [16] and by Hey [17, 18], which discuss propertiesof the momentum space hydrogen functions. In [5, SectionIV], I presented a different and remarkably simple derivationof the Fourier transform of a bound state hydrogen eigenfunc-tion and of related functions which will play a major role inSection VI.In Section II, basic properties of the generalized Laguerrepolynomials and of bound state hydrogen eigenfunctions arereviewed. As discussed in Section III, bound state hydrogeneigenfunctions are in contrast to several other similar functionsets not complete in the Hilbert space L ( R ) of square inte-grable functions. This makes bound state hydrogen eigenfunc-tions useless in expansions. Apparently, Y¨ukc¸ ¨u and Y¨ukc¸ ¨uare unaware of this well known and very consequential fact.The explicit expressions derived by Y¨ukc¸ ¨u and Y¨ukc¸ ¨u [2,Eqs. (15) and (16)] are less useful than those mentioned above(compare the discussion in Section V). This is a direct conse-quence of their derivation: Y¨ukc¸ ¨u and Y¨ukc¸ ¨u expressed gen-eralized Laguerre polynomials in terms of powers. Superfi-cially, this looks quite natural, but actually it is a bad idea.Section VI shows how their approach can be improved sub-stantially by expanding generalized Laguerre polynomials interms of better suited alternative function sets, the so-calledreduced Bessel functions. II. GENERALIZED LAGUERRE POLYNOMIALS ANDBOUND-STATE HYDROGEN EIGENFUNCTIONS
The generalized Laguerre polynomials L ( α ) n ( z ) with ℜ ( α ) > − and n ∈ N are the classical orthogonal poly-nomials associated with the integration interval [0 , ∞ ) andthe weight function w ( z ) = z α exp( − z ) . They are of con-siderable importance in mathematics and also in theoreticalphysics. There is a detailed literature which is far too exten-sive to be cited here. Those interested in the historical devel-opment with a special emphasis on quantum physics shouldconsult an article by Mawhin and Ronveaux [19]. GeneralizedLaguerre polynomials also played a major role in my own re-search [5, 20–24].It is recommendable to use the modern mathematical defi-nition of the generalized Laguerre polynomials L ( α ) n ( z ) with n ∈ N and α, z ∈ C , which are defined either via their Ro-drigues’ relationship [25, Eq. (18.5.5) and Table 18.5.1] oras a terminating confluent hypergeometric series F [25, Eq.(18.5.12)]: L ( α ) n ( z ) = z − α e z n ! d n d z n (cid:2) e − z z n + α (cid:3) (1) = ( α + 1) n n ! F ( − n ; α + 1; z ) . (2)Further details can be found in books on special functions.Dating back from the early days of quantum mechanics, anantiquated notation is still frequently used mainly in atomictheory. For example, Bethe and Salpeter [10, Eq. (3.5)] in-troduced so-called associated Laguerre functions (cid:2) L mn ( z ) (cid:3) BS with n, m ∈ N via the Rodrigues-type relationships (cid:2) L mn ( z ) (cid:3) BS = d m d z m (cid:2) L n ( z ) (cid:3) BS , (3a) (cid:2) L n ( z ) (cid:3) BS = e z d n d z n (cid:2) e − z z n (cid:3) . (3b)This convention is also used in the books by Condon andShortley [26, Eqs. (6) and (9) on p. 115] and Condon andOdabas¸i [27, Eq. (2) on p. 189].Generalized Laguerre polynomials with integral superscript α = m ∈ N and the associated Laguerre functions (3) areconnected via L ( m ) n ( z ) = ( − m ( n + m )! (cid:2) L mn + m ( z ) (cid:3) BS , m, n ∈ N . (4)The notation for associated Laguerre functions is less intu-itive than the notation for the generalized Laguerre polyno-mials, whose subscript n corresponds to the polynomial de-gree and whose superscript α characterizes the weight func-tion w ( z ) = z α exp( − z ) . The worst drawback of the func-tions (3) is that they cannot express generalized Laguerre poly-nomials L ( α ) n with non-integral superscripts α which also oc-cur in quantum physics. The eigenfunctions Ω mn,ℓ ( β, r ) of theHamiltonian β − ∇ − β r of the three-dimensional isotropicharmonic oscillator contain generalized Laguerre polynomialsin r with half-integral superscripts (see for example [5, Eq.(5.4)] and references therein). Similarly, the eigenfunctionsof the Dirac equation for the hydrogen atom contain general-ized Laguerre polynomials with in general non-integral super-scripts [13, Eqs. (9.84) and (9.85)].If the modern mathematical notation is used, the bound-state eigenfunctions of a hydrogenlike ion with nuclear charge Z in spherical polar coordinates is essentially the product ofan exponential and a generalized Laguerre polynomial, bothdepending on r , and a regular solid harmonic Y mℓ ( r ) = r ℓ Y mℓ ( θ, φ ) (see for example [12, Eqs. (7.4.41) - (7.4.43)] or[13, Eqs. (9.2) and (9.10)]): W mn,ℓ ( Z, r ) = (cid:18) Zn (cid:19) / (cid:20) ( n − ℓ − n ( n + ℓ )! (cid:21) / × e − Zr/n L (2 ℓ +1) n − ℓ − (2 Zr/n ) Y mℓ (2 Z r /n ) ,n ∈ N , ℓ ∈ N ≤ n − , − ℓ ≤ m ≤ ℓ . (5)Y¨ukc¸ ¨u and Y¨ukc¸ ¨u [2] define the radial part of the bound-state eigenfunctions (5) via their Eq. (3), which is inconsis-tent with their definition of the generalized Laguerre polyno-mials via their Eq. (11). It can be shown that their Eq. (11)is equivalent to Eq. (2) which implies that Y¨ukc¸ ¨u and Y¨ukc¸ ¨ualso use the modern mathematical notation. In addition, theirRef. [26] for their Eq. (11) is incorrect. The so-called Bate-man Manuscript Project [28–32] was named to honor HarryBateman who had died in 1946, i.e., long before these bookshad been completed. Thus, the correct reference for Eq. (11)of Y¨ukc¸ ¨u and Y¨ukc¸ ¨u [2] would be [29, Eq. (7) on p. 188].
III. INCOMPLETENESS OF THE BOUND-STATEHYDROGEN EIGENFUNCTIONS
Expansions of a given function in terms of suitable func-tion sets are among the most useful techniques of mathemati-cal physics. This approach requires that the function set beingused is complete and preferably also orthogonal in the corre-sponding Hilbert space. As for example discussed in [23] or in[33], non-orthogonal expansions can easily have pathologicalproperties.Bound-state hydrogenic eigenfunctions (5) are orthonormalwith respect to an integration over the whole R , Z (cid:2) W mn,ℓ ( Z, r ) (cid:3) ∗ W m ′ n ′ ,ℓ ′ ( Z, r ) d r = δ nn ′ δ ℓℓ ′ δ mm ′ , (6)but they are not complete in the Hilbert space L ( R ) = n f : R → C (cid:12)(cid:12)(cid:12) Z | f ( r ) | d r < ∞ o (7)of square integrable functions without the inclusion of thetechnically very difficult continuum eigenfunctions, describedfor instance in [10, pp. 21 - 25], in [25, Chapter 33 CoulombFunctions] or in the recent article [34]. Y¨ukc¸ ¨u and Y¨ukc¸ ¨u areapparently not aware of this incompleteness.In the literature, this incompleteness, which was first de-scribed in 1928 by Hylleraas [35, p. 469], is sometimes over-looked – often with catastrophic consequences. For exam-ple, Y¨ukc¸ ¨u and Y¨ukc¸ ¨u cited as their Ref. [4] an article byYamaguchi [36] in order to demonstrate the usefulness ofbound-state hydrogen eigenfunctions in expansions. However,Yamaguchi’s article had been severely criticized in [21] forsimply neglecting the troublesome continuum eigenfunctions.Already in 1955, Shull and L¨owdin [37] had emphasized theimportance of the continuum eigenfunctions and tried to esti-mate the magnitude of the error due to their omission.At first sight, this incompleteness may seem surprisingsince the completeness of the generalized Laguerre polyno-mials L ( α ) n ( z ) in the weighted Hilbert space L − z z α (cid:0) [0 , ∞ ) (cid:1) = n f : C → C (cid:12)(cid:12)(cid:12) Z ∞ e − z z α | f ( z ) | d z < ∞ o (8)is a classic result of mathematical analysis (see for examplethe books by Higgins [38, p. 33], Sansone [39, pp. 349 - 351],Szeg¨o [40, pp. 108 - 110], or Tricomi [41, pp. 235 - 238]).Thus, every function f ∈ L − z z α (cid:0) [0 , ∞ ) (cid:1) can be expressedby a Laguerre series f ( z ) = ∞ X n =0 λ ( α ) n L ( α ) n ( z ) , (9a) λ ( α ) n = n !Γ( α + n + 1) Z ∞ z α e − z L ( α ) n ( z ) f ( z ) d z , (9b)which converges in the mean with respect to the norm of theHilbert space L − z z α (cid:0) [0 , ∞ ) (cid:1) . For a condensed discussion ofLaguerre expansions, see [22, Section 2].How can the incompleteness of the bound-state hydro-gen eigenfunctions (5) be explained? The culprit is their n -dependent scaling parameter Z/n . Fock [42, Eq. (6.17) on p.200] showed that the confluent hypergeometric function F (cid:0) − n + ℓ + 1; 2 ℓ + 2; 2 Zr/n (cid:1) = n − ℓ − X ν =0 ( − n + ℓ + 1) ν (2 ℓ + 2) ν [2 Zr/n ] ν ν ! (10)occurring in Eq. (5) can in the limit n → ∞ be representedby a Bessel function J ℓ +1 (cid:16) √ Zr (cid:17) of the first kind, whichis an oscillatory function that decays too slowly to be squareintegrable (compare also [25, Eq. (18.11.6)]). In the limit n → ∞ , the exponential exp( − Zr/n ) in Eq. (5) loses itsexponential decay as r → ∞ . Consequently, the bound statehydrogen eigenfunctions (5) become oscillatory as n → ∞ ,which means that they are no longer square integrable. In-stead, they belong to the continuous spectrum. Thus, the so-called bound-state eigenfunctions are no longer bound-statefunctions if the principal quantum number n becomes verylarge. This implies that the bound-state eigenfunctions cannotform a basis for the Hilbert space L ( R ) of square integrablefunctions (compare [42, text following Eq. (6.19) on p. 201]).Because of the incompleteness of the bound-state hydrogeneigenfunction, it is now common to use in expansions alter-native function sets also based on the generalized Laguerrepolynomials that possess more convenient completeness prop-erties. Closely related to the bound-state hydrogenic eigen-functions are the so-called Coulomb Sturmians or Sturmianswhich were already used in 1928 by Hylleraas [35, Eq. (25)on p. 478]: Ψ mn,ℓ ( β, r ) = (2 β ) / (cid:20) ( n − ℓ − n ( n + ℓ )! (cid:21) / × e − βr L (2 ℓ +1) n − ℓ − (2 βr ) Y mℓ (2 β r ) . (11) Here, the notation of [5, Eq. (4.6)] is used. We obtain bound-state hydrogen eigenfunctions (5) with a correct normalizationfactor if we make in Eq. (11) the substitution β Z/n (com-pare the discussion following [5, Eq. (4.12)]): Ψ mn,ℓ ( Z/n, r ) = W mn,ℓ ( Z, r ) . (12)This is a non-trivial result. Sturmians are complete and or-thonormal in the in the Sobolev space W (1)2 ( R ) (for the def-inition of Sobolev spaces plus further references, see [5, Sec-tion II]), whereas bound state hydrogen functions are orthonor-mal but incomplete in the Hilbert space L ( R ) .Sturmians occur in the context of Fock’s treatment of thehydrogen atom [4], albeit in a somewhat disguised form (com-pare [5, Section VI]). There is a classic review by Rotenberg[43]. A fairly detailed discussion of their properties was givenby Novosadov [44]. Sturmians also play a major role in booksby Avery [6, 7], Avery and Avery [8], and Avery, Rettrup, andAvery [9]. We used Sturmians for the construction for an addi-tion theorem of the Yukawa potential [45] with the help ofweakly convergent orthogonal and biorthogonal expansionsfor the plane wave introduced in [5, Section III].Lambda functions were introduced already in 1929 byHylleraas [46, Footnote ∗ on p. 349], and later by Shull andL¨owdin [37] and by L¨owdin and Shull [47, Eq. (46)]: Λ mn,ℓ ( β, r ) = (2 β ) / (cid:20) ( n − ℓ − n + ℓ + 1)! (cid:21) / × e − βr L (2 ℓ +2) n − ℓ − (2 βr ) Y mℓ (2 β r ) . (13)Here, the notation of [5, Eq. (4.4)] is used.The use of Lambda functions in electronic structure the-ory was suggested by Kutzelnigg [48] and Smeyers [49] in1963 and 1966, respectively. Filter and Steinborn [50] usedthem for the derivation of one-range addition theorems of ex-ponentially decaying functions, and I used both Sturmians andLambda functions for the construction of weakly convergentexpansions of a plane wave [5].Both Sturmians and Lambda functions defined by Eqs. (11)and (13) have a fixed scaling parameter β > that does not de-pend on the principal quantum number n . Consequently, thesefunctions are orthogonal and complete in suitable Hilbert andSobolev spaces. A detailed discussion of the mathematicalproperties of the functions Ψ mn,ℓ ( β, r ) and Λ mn,ℓ ( β, r ) wasgiven in [5, Section IV] or in [23, Section 2]. IV. THE WORK OF PODOLSKY AND PAULING
The Fourier transform of an irreducible spherical tensor ofintegral rank yields a Hankel-type radial integral multiplied bya spherical harmonic if the so-called Rayleigh expansion of aplane wave (compare for instance [12, p. 442]) is used: e ± i x · y = 4 π ∞ X ℓ =0 ( ± i) ℓ j ℓ ( xy ) × ℓ X m = − ℓ (cid:2) Y mℓ ( x /x ) (cid:3) ∗ Y mℓ ( y /y ) , x , y ∈ R . (14)With the help of the orthonormality of the spherical harmonicsand the definition of the spherical Bessel functions j ℓ ( xy ) (seefor example [25, Eq. (10.47.3)]), we obtain the following ex-pression for the Fourier transformation of a Sturmian function(11) without normalization factor and with fixed β > : (2 π ) − / Z e i p · r e − βr r ℓ L (2 ℓ +1) n (2 βr ) Y mℓ ( r /r ) d r = ( − i) ℓ p − / Y mℓ ( p /p ) × Z ∞ r ℓ +3 / e − βr J ℓ +1 / ( pr ) L (2 ℓ +1) n (2 βr ) d r . (15)For a closed form expression of the Hankel-type radial integralin Eq. (15), we need an explicit expression for the integral I ( α,µ,ν ) n ( a, b ) = Z ∞ y µ e − ay J ν ( by ) L ( α ) n (2 ay ) d y . (16)In 1929, when Podolsky and Pauling [1] tried to derive anexpression for the Fourier transform of a bound-state hydro-gen eigenfunction, no explicit expression for this integral wasknown. Even today, I could not find the required expressionin the usual books on special function theory.Podolsky and Pauling [1, Eq. (6)] found a very elegant so-lution to this problem. Their starting point was the generatingfunction [51, p. 242] exp (cid:16) xtt − (cid:17) (1 − t ) α +1 = ∞ X n =0 L ( α ) n ( x ) t n , | t | < . (17)Inserting this generating function of the generalized Laguerrepolynomials into the radial integral in Eq. (15) yields: ∞ X n =0 t n Z ∞ r ℓ +3 / e − βr J ℓ +1 / ( pr ) L (2 ℓ +1) n (2 βr ) d r = (1 − t ) − ℓ − × Z ∞ e − βr t − t r ℓ +3 / J ℓ +1 / ( pr ) d r . (18)The radial integral on the right-hand side can be expressedin closed form. We use [52, Eq. (2) on p. 385] Z ∞ e − ay J ν ( by ) y µ − d y = ( b/ ν Γ( µ + ν ) a µ + ν Γ( ν + 1) × F (cid:18) µ + ν , µ + ν + 12 ; ν + 1; − b a (cid:19) , ℜ ( a ± i b ) > , (19)to obtain Z ∞ e − βr t − t r ℓ +3 / J ℓ +1 / ( pr ) d r = (2 ℓ + 2)!Γ( ℓ + 3 /
2) ( p/ ℓ +1 / (cid:2) β (1 + t ) / (1 − t ) (cid:3) ℓ +3 × F (cid:18) ℓ + 3 / , ℓ + 2; ℓ + 3 / − p β (1 − t ) (1 + t ) (cid:19) . (20) This Gaussian hypergeometric series F is actually a bi-nomial series F ( ℓ + 2; z ) = P ∞ m =0 ( ℓ + 2) m z m /m ! =(1 − z ) − ℓ − with z = − p (1 − t ) / [ β [1 + t ] ] [25, Eq.(15.4.6)]. Thus, we obtain for the right-hand side of Eq. (18): (1 − t ) − ℓ − Z ∞ e − βr t − t r ℓ +3 / J ℓ +1 / ( pr ) d r = (2 ℓ + 2)!Γ( ℓ + 3 /
2) ( p/ ℓ +1 / β (1 − t ) (cid:2) β (1 + t ) + p (1 − t ) (cid:3) ℓ +2 (21)The denominator can be simplified further, using β (1 + t ) + p (1 − t ) = (cid:0) β + p (cid:1)(cid:8) (cid:2) (cid:0) β − p (cid:1)(cid:3) (cid:2) β + p (cid:3) t + t (cid:9) ,yielding (1 − t ) − ℓ − Z ∞ e − βr t − t r ℓ +3 / J ℓ +1 / ( pr ) d r = (2 ℓ + 2)!Γ( ℓ + 3 / × ( p/ ℓ +1 / β (1 − t ) "(cid:0) β + p (cid:1) ( (cid:0) β − p (cid:1) β + p t + t ) ℓ +2 . (22)The rational function on the right-hand side closely resemblesthe generating function [51, p. 222] (cid:0) − xt + t (cid:1) − λ = ∞ X n =0 C λn ( x ) t n , | t | < , (23)of the Gegenbauer polynomials. Podolsky and Pauling onlyhad apply the differential operator t − λ [ ∂/∂t ] t λ to Eq. (23).This yields the following modified generating function of theGegenbauer polynomials (compare [1, Eq. (25 )]), − t (cid:0) − xt + t (cid:1) λ +1 = ∞ X n =0 λ + nλ C λn ( x ) t n , (24)which I could not find in the usual books on special functiontheory. The rational function on the right-hand side of Eq. (22)is of the same type as the left-hand side of this modified gen-erating function. If we make in Eq. (24) the substitutions x ( p − β ) / ( p + β ) and λ ℓ + 1 , we obtain thefollowing expansion in terms of Gegenbauer polynomials: (cid:0) − t (cid:1)(cid:30)(cid:26) − p − β p + β t + t (cid:27) ℓ +2 = ∞ X n =0 n + ℓ + 1 ℓ + 1 C ℓ +1 n (cid:18) p − β p + β (cid:19) t n . (25)Inserting this into Eq. (22) yields: (1 − t ) − ℓ − Z ∞ e − βr t − t r ℓ +3 / J ℓ +1 / ( pr ) d r = ( p/ ℓ +1 / (2 ℓ + 2)!( ℓ + 1)Γ( ℓ + 3 / β × ∞ X n =0 n + ℓ + 1 (cid:2) p + β (cid:3) ℓ +2 C ℓ +1 n (cid:18) p − β p + β (cid:19) t n . (26)Thus, we finally obtain the following explicit expression forthe Fourier transform of an unnormalized Sturmian: (2 π ) − / Z e i p · r e − βr L (2 ℓ +1) n − ℓ − (2 βr ) Y mℓ (2 β r ) d r = (2 /π ) / ℓ +1 ℓ ! β ℓ +1 n (cid:2) p + β (cid:3) ℓ +2 × C ℓ +1 n − ℓ − (cid:18) p − β p + β (cid:19) Y mℓ ( − i p ) . (27)To obtain the Fourier transform of a normalized Sturmian de-fined by Eq. (11), we multiply Eq. (27) by the normaliza-tion factor (2 β ) / [( n − ℓ − / [2 n ( n + ℓ )!]] / , yielding[5, Eq. (4.24)]: Ψ mn,ℓ ( β, p ) = (2 π ) − / Z e i p · r Ψ mn,ℓ ( β, r ) d r = 2 ℓ ℓ ! (cid:20) βn ( n − ℓ − π ( n + ℓ )! (cid:21) / (cid:20) βp + β (cid:21) ℓ +2 × C ℓ +1 n − ℓ − (cid:18) p − β p + β (cid:19) Y mℓ ( − i p ) . (28)To obtain the Fourier transform of a bound-state hydrogeneigenfunction, we only have to use Eq. (12) and make the sub-stitution β Z/n . Thus, we obtain [1, Eq. (28)]: W mn,ℓ ( Z, p ) = (2 π ) − / Z e i p · r W mn,ℓ ( Z, r ) d r = 2 ℓ ℓ ! (cid:20) Z ( n − ℓ − π ( n + ℓ )! (cid:21) / (cid:20) Znn p + Z (cid:21) ℓ +2 × C ℓ +1 n − ℓ − (cid:18) n p − Z n p + Z (cid:19) Y mℓ ( − i p ) . (29)This Fourier transformation was in principle also derived byRotenberg [43, Eq. (26) on p. 241] in disguised form. How-ever, Rotenberg’s results are misleading because of an unfor-tunate definition of the Sturmians (compare [5, p. 283]).If we compare Eq. (29) with formulas published by otherauthors, we find some discrepancies. In the formula givenby Podolsky and Pauling [1, Eq. (28)], a phase factor ( − i) ℓ is missing. The same error was reproduced by Bethe andSalpeter [10, Eq. (8.8)]. The formula given by Englefield [11,Eqs. (5.5) and (5.6)] differs from Eq. (29) by a phase factor ( − m . Finally, in the expression given by Biedenharn andLouck [12, Eq. (7.4.69)] a factor π − / is missing.Kaijser and Smith [53, pp. 50 - 52] showed that the gen-erating function approach of Podolsky and Pauling can be ex-tended to the Fourier transform of a Lambda function definedby Eq. (13). However, the approach of Podolsky and Paul-ing [1] and Kaijser and Smith [53] requires considerable ma-nipulative skills. In Section VI, I will show how the Fouriertransforms of bound-state hydrogen eigenfunctions, Sturmi-ans, and Lambda functions and of other Laguerre-type func-tions can be constructed in an almost trivially simple way byexpanding generalized Laguerre polynomials in terms of so-called reduced Bessel functions (compare [5, Section IV]). V. THE WORK OF Y ¨UKC¸ ¨U AND Y ¨UKC¸ ¨U
Podolsky and Pauling [1] faced the problem that no sim-ple closed form expression for the Hankel-type integral inEq. (15) was known. They solved this problem by com-puting instead the Fourier transform of the generating func-tion (17), which leads to the comparatively simple and ex-plicitly known Hankel-type integral in Eq. (18). In this way,Podolsky and Pauling only had to perform a series expansionof the radial integral in Eq. (18) to derive the explicit expres-sion (29) for the Fourier transform of a bound state hydrogeneigenfunction.Y¨ukc¸ ¨u and Y¨ukc¸ ¨u [2], who were apparently unaware of thework by Podolsky and Pauling [1] or of the whole extensiveliterature on this topic, proceeded differently. They utilizedthe fact that a generalized Laguerre polynomial L ( α ) n ( z ) is ac-cording to Eq. (2) a polynomial of degree n in z . Thus, thegeneralized Laguerre polynomial L (2 ℓ +1) n − ℓ − (2 βr ) occurring inEq. (11) can be expressed as a sum of powers: L (2 ℓ +1) n − ℓ − (2 βr )= ( n + ℓ )!( n − ℓ − n − ℓ − X ν =0 ( − n + ℓ + 1) ν (2 ℓ + ν + 1)! (2 βr ) ν ν ! . (30)To achieve what they believe to be a further simplification,Y¨ukc¸ ¨u and Y¨ukc¸ ¨u [2, Eqs. (10) - (12)] combined Eq. (30) withthe Laguerre multiplication theorem [51, p. 249] L ( α ) n ( zx ) = n X m =0 (cid:18) n + αm (cid:19) (1 − z ) m z m − n L ( α ) n − m ( x ) . (31)However, the combination of Eqs. (30) and (31) leads to thesame Hankel-type integrals as the direct use of Eq. (30). Thus,this combination accomplishes nothing and only introducesa completely useless additional inner sum. Therefore, I willonly consider the direct use of Eq. (30).In 1930, Slater [54] introduced the so-called Slater-typefunctions, which had an enormous impact on atomic elec-tronic structure theory and which in unnormalized form areexpressed as follows: χ mn,ℓ ( a, r ) = ( αr ) n − e − αr Y mℓ ( θ, φ )= ( αr ) n − ℓ − e − αr Y mℓ ( α r ) , α > . (32)I always tacitly assume that the principal quantum number n is a positive integer n ∈ N satisfying n − ℓ ≥ .With the help of Slater-type functions, an unnormalizedSturmian function (11) with fixed β > can be expressedas follows: e − βr L (2 ℓ +1) n − ℓ − (2 βr ) Y mℓ (2 β r ) = ( n + ℓ )!( n − ℓ − × n − ℓ − X ν =0 ( − n + ℓ + 1) ν (2 ℓ + ν + 1)! 2 ν ν ! χ mν + ℓ +1 ,ℓ ( β, r ) . (33)The idea of expressing functions based on the generalizedLaguerre polynomial by finite sums of Slater-type functionsis not new. To the best of my knowledge, it was introducedby Smeyers [49] in 1966, who expressed Lambda functionsdefined by Eq. (13) as linear combinations of Slater-type func-tions. Smeyers constructed in this way one-range addition the-orems of Slater-type functions, which were expansion in termsof Lambda functions. Thus, their expansion coefficients areoverlap integrals [49, Section 3]. In 1978, Guseinov [55, Eqs.(6) - (8)] adopted Smeyers’ approach and consistently used itin his countless later publications, without ever giving creditto Smeyers [49].Smeyers’ approach is undoubtedly very simple. Neverthe-less, it is not good. In Eq. (30) there are strictly alternatingsigns. Therefore, in sums of the type of Eq. (33), which inheritthe alternating signs from Eq. (30), numerical instabilities areto be expected in the case of larger summation indices. Thishad already been emphasized in 1982 by Trivedi and Stein-born [56, pp. 116 - 117]. For a more detailed discussion plusadditional references, see [23, pp. 32 - 34].Fourier transformation is a linear operation. Consequently,Eq. (33) implies that the Fourier transformation of a Sturmian– or of any of the various other functions based on gener-alized Laguerre polynomials – can be expressed as a finitelinear combination of Fourier transforms of Slater-type func-tions with integral principal quantum numbers (compare [2,Eq. (12)]).There is an extensive literature on Fourier transforms ofSlater-type functions. I am aware of articles by Geller [57, 58],Silverstone [59, 60], Edwards, Gottlieb, and Doddrell [61],Henneker and Cade [62], Kaijser and Smith [53]. Wenigerand Steinborn [20], Niukkanen [63], Belki´c and Taylor [64],and by Akdemir [65]. In addition, there is a Wikipedia arti-cle [66], whose principal reference is the article by Belki´c andTaylor [64]. Y¨ukc¸ ¨u and Y¨ukc¸ ¨u [2] only mentioned Niukkanen[63] as their Ref. [8].The Rayleigh expansion (14) leads to an expression of theFourier transform of a Slater-type function as a Hankel-typeradial integral: χ mn,ℓ ( α, p ) = (2 π ) − / Z e − i p · r χ mn,ℓ ( α, r ) d r = α n − ( − i) ℓ Y mℓ ( p /p ) × p − / Z ∞ e − αr r n +1 / J ℓ +1 / ( pr ) d r . (34)This Hankel-type integral is a special case of the one inEq. (19). Thus, we immediately obtain [20, Eq. (3.15)] χ mn,ℓ ( α, p ) = ( n + ℓ + 1)! α ℓ +3 (2 π ) / (1 / ℓ +1 Y mℓ ( − i p / × F (cid:18) n + ℓ + 22 , n + ℓ + 32 ; ℓ + 32 ; − p α (cid:19) , (35)which corresponds to [2, Eqs. (14) and (16)].For the derivation of Watson’s hypergeometric representa-tion (19), one only has to insert the power series J ν ( z ) = (cid:2) ( z/ ν / Γ( ν + 1) (cid:3) F ( ν + 1; − z / [25, Eq. (10.16.9)] intothe integral, followed by an interchange of summation andterm-wise integration. In the final step, the Pochhammer du-plication formula [25, Eq. (5.2.8)] is to be used. Watson’s derivation can be modified easily. If we insteaduse J ν ( z ) = (cid:2) ( z/ ν e ∓ i z / Γ( ν + 1) (cid:3) F ( ν + 1 /
2; 2 ν +1; ± z ) [25, Eq. (10.16.5)], we obtain an alternative repre-sentation which involves a F with complex argument: Z ∞ e − at J ν ( bt ) t µ − d t = ( b/ ν ( a ± i b ) µ + ν Γ( µ + ν )Γ( ν + 1) × F (cid:18) ν + 12 , µ + ν ; 2 ν + 1; ± ba ± i b (cid:19) . (36)This yields the following Fourier transformation: χ mn,ℓ ( α, p ) = ( n + ℓ + 1)!(2 π ) / (1 / ℓ +1 α n − ( α ± i p ) n + ℓ +2 Y mℓ ( − i p / × F (cid:18) n + ℓ + 2 , ℓ + 1; 2 ℓ + 2; ± pα ± i p (cid:19) . (37)A slightly less general result had been obtained by Belki´c andTaylor [64, Eq. (15)]. They used only the upper signs in theexpression involving the F given above. In this way, Belki´cand Taylor [64, Eq. (15)] obtained only the upper signs on theright-hand side of Eq. (37).The Hankel-type integral in Eq. (36) is real if µ, ν, a, p ∈ R .Thus, the right-hand side of Eq. (36) also has to be real, orequivalently, it has to be equal to its complex conjugate. Thisimplies: F (cid:18) ν + 12 , µ + ν ; 2 ν + 1; ± ba ± i b (cid:19) = (cid:18) a ± i ba ∓ i b (cid:19) µ + ν × F (cid:18) ν + 12 , µ + ν ; 2 ν + 1; ∓ ba ∓ i b (cid:19) . (38)Analogous symmetries also exist in Eq. (37) and in all otherexpressions of that kind derived later.The hypergeometric series F in Eqs. (35) and (37) do notconverge for all p ≥ . We either have | − p /α | → ∞ as p → ∞ , or | ± p/ ( α ± i p ) | → as p → ∞ . Thus, Eqs. (35)and (37) are not sufficient for computational purposes. Theyare, however, convenient starting points for the constructionof alternative expressions with better numerical properties. Inthe case of the F in Eq. (35), this had already been empha-sized in the first edition of Watson’s classic book [52, Eq. (3)on p. 385] which appeared in 1922.For the construction of analytic continuation formulas, itmakes sense to use the highly developed transformation the-ory of the Gaussian hypergeometric function F as an order-ing principle (see for example [51, pp. 47 - 51] or [25, § Hypergeometric Func-tions in Mathematics and Theoretical Physics [67].The simplest transformations of a F are the Euler andPfaff transformations (see for example [51, p. 47]): F ( a, b ; c ; z )= (1 − z ) c − a − b F ( c − a, c − b ; c ; z ) (39) = (1 − z ) − a F (cid:0) a, c − b ; c ; z/ ( z − (cid:1) (40) = (1 − z ) − b F (cid:0) c − a, b ; c ; z/ ( z − (cid:1) . (41)The application the Euler transformation (39) to the F s inEqs. (35) and (37) yields [20, Eq. (3.11)] χ mn,ℓ ( α, p ) = ( n + ℓ + 1)!(2 π ) / (1 / ℓ +1 Y mℓ ( − i p / × α n − ℓ − [ α + p ] n +1 × F (cid:18) ℓ − n , ℓ − n + 12 ; ℓ + 32 ; − p α (cid:19) (42)and χ mn,ℓ ( α, p ) = ( n + ℓ + 1)!(2 π ) / (1 / ℓ +1 Y mℓ ( − i p / × α n − ( α ± i p ) ℓ +1 ( α ∓ i p ) n +12 F (cid:18) ℓ − n, ℓ + 1; 2 ℓ + 2; ± pα ± i p (cid:19) . (43)Since we assume n ∈ N , ℓ ∈ N , n − ℓ − ≥ , n − ℓ and either ( ℓ − n ) / or ( ℓ − n + 1) / are positive integers. Accordingly,the F s in Eqs. (42) and (43) terminate, which representsa substantial improvement compared to the non-terminating F s in Eqs. (35) and (37). Both Eqs. (42) and (43) allowa convenient evaluation of χ mn,ℓ ( α, p ) for all p ∈ R . Forrecurrence formulas of the Gaussian hypergeometric function F ( a, b ; c ; z ) , where two or three of the parameters a , b , and c change simultaneously , see [68, Appendix C].We can also employ the Pfaff transformations (40) and (41).In the case of Eq. (35), this yields hypergeometric series withargument p / ( α + p ) that either terminate or converge forall p ≥ [20, Eqs. (3.16) and (3.17)]. In the case of Eq. (43),we only obtain complex conjugates of known radial parts.But this is not yet the end of the story. By systematicallyexploiting the known transformation properties of the Gaus-sian hypergeometric function F , many other terminating ornon-terminating expressions can be derived. For example, wecould also use one of the linear transformations that accom-plish the variable transformations z − z , z /z , z / (1 − z ) , and z − /z , respectively, by expressinga given F in terms of two other F s (see for example [51,pp. 47 - 49]). Normally, these transformations lead to compar-atively complicated expressions which can safely be ignored.An exception is the following expression obtained by a trans-formation z / (1 − z ) [20, Eqs. (3.19) and (3.20)]: χ mn,ℓ ( α, p ) = ( π/ / ( n + ℓ + 1)! Y mℓ ( − i p / × " α n − [ α + p ] − ( n + ℓ +2) / Γ (cid:0) [ n + ℓ + 3] / (cid:1) Γ (cid:0) [ ℓ − n + 1] / (cid:1) × F (cid:18) n + ℓ + 32 , ℓ − n α α + p (cid:19) − α n [ α + p ] − ( n + ℓ +3) / Γ (cid:0) [ n + ℓ + 2] / (cid:1) Γ (cid:0) [ ℓ − n ] / (cid:1) × F (cid:18) n + ℓ + 32 , ℓ − n + 12 ; 32 ; α α + p (cid:19) . (44)This expression is simpler than it looks. If n − ℓ is even,the second part of the right-hand side vanishes because of thegamma function Γ (cid:0) [ ℓ − n ] / (cid:1) , and if n − ℓ is odd, the first partvanishes because of the gamma function Γ (cid:0) [ ℓ − n + 1] / (cid:1) .In addition to linear transformations, a Gaussian hypergeo-metric function F may also satisfy so-called quadratic trans-formations (see for example [51, pp. 49 - 51] or [25, § F ( a, b ; c ; z ) with completely arbitrary parameters a , b ,and c . They only exists for special values of the parameters a , b , and c [25, Table 15.8.1].The hypergeometric series in Eq. (35) is of the general type F ( a, a + 1 / c ; z ) . This suggests the application of the fol-lowing quadratic transformations [51, p. 50] to this F : F (cid:18) a, a + 12 ; c ; z (cid:19) = (1 − z ) − a F (cid:18) a, c − a − c ; √ − z − √ − z (cid:19) (45) = (cid:0) ± √ z (cid:1) − a F (cid:18) a, c −
12 ; 2 c − ± √ z ± √ z (cid:19) (46) = (cid:18) √ − z (cid:19) − a × F (cid:18) a, a − c + 1; c ; 1 − √ − z √ − z (cid:19) (47)Application of Eqs. (45) – (47) to the F in Eq. (35) yieldsthe following alternative expressions: χ mn,ℓ ( α, p )= ( n + ℓ + 1)!(2 π ) / (1 / ℓ +1 α n − [ α + p ] ( n + ℓ +2) / Y mℓ ( − i p / × F n + ℓ + 2 , ℓ − n ; ℓ + 32 ; p α + p − α p α + p ! (48) = ( n + ℓ + 1)!(2 π ) / (1 / ℓ +1 α n − ( α ± i p ) n + ℓ +2 Y mℓ ( − i p / × F (cid:18) n + ℓ + 2 , ℓ + 1; 2 ℓ + 2; ± pα ± i p (cid:19) (49) = ( n + ℓ + 1)!(2 π ) / (1 / ℓ +1 n + ℓ +2 α n − [ α + p ] ( n + ℓ +2) / Y mℓ ( − i p / × F n + ℓ + 2 , n + 32 ; ℓ + 32 ; α − p α + p α + p α + p ! . (50)Equations (37) and (49) are identical. The derivation of Eq. (49) shows that the quadratic transformation (46) can cre-ate a representation containing a F with complex argumentfrom a F with real argument. Consequently, the derivationof the complex expression (36) for the Hankel-type integral inEq. (19) by directly evaluating the integral is – strictly speak-ing – superfluous. Applying the quadratic transformation (46)to the F in Eq. (19) would have done the job.Only the F in Eq. (48) terminates. As a remedy, we can apply the Euler transformation (39) to the non-terminating F s inEqs. (49) and (50), yielding χ mn,ℓ ( α, p )= ( n + ℓ + 1)!(2 π ) / (1 / ℓ +1 α n − ( α ± i p ) ℓ +1 ( α ∓ i p ) n +1 Y mℓ ( − i p / F (cid:18) ℓ − n, ℓ + 1; 2 ℓ + 2; ± pα ± i p (cid:19) (51) = ( n + ℓ + 1)!(2 π ) / (1 / ℓ +1 α n − n − ℓ h α + p α + p i n +2 [ α + p ] (3 n + ℓ +4) / Y mℓ ( − i p / F ℓ − n, − n −
12 ; ℓ + 32 ; α − p α + p α + p α + p ! . (52)The terminating F in Eq. (48) can be expressed as aGegenbauer polynomial via [51, p. 220] C λn ( x ) = (2 λ ) n n ! F (cid:18) − n, n + 2 λ ; λ + 12 ; 1 − x (cid:19) , (53)yielding χ mn,ℓ ( α, p ) = ( n + ℓ + 1)! ( n − ℓ )!(2 π ) / (1 / ℓ +1 (2 ℓ + 2) n − ℓ Y mℓ ( − i p / × α n − [ α + p ] n + ℓ +1 C ℓ +1 n − ℓ α p α + p ! . (54)The Gegenbauer polynomial representation (54) correspondsto the second representation given by Y¨ukc¸ ¨u and Y¨ukc¸ ¨u [2] intheir Eqs. (13) and (15). As their source, Y¨ukc¸ ¨u and Y¨ukc¸ ¨u[2] give the book by Gradshteyn and Rhyzhik [69] as their Ref.[25], without specifying a page or equation number. Unfortu-nately, I was not able to find the corresponding expression inthe book by Gradshteyn and Rhyzhik [69].In earlier articles by Yavuz, Y¨ukc¸ ¨u, ¨Oztekin, Yılmaz, andD¨ond¨ur [70, Eq. (12)] and Y¨ukc¸ ¨u [71, Eq. (32)], the Gegen-bauer polynomial representation (54) had been attributedto Guseinov [72]. Google Scholar gave me the title ofGuseinov’s article, but I was not able to obtain a copy. Per-sonal contacts to the Wuhan Institute of Physics and Mathe-matics of the Chinese Academy of Sciences could not help,either.But Guseinov was not the first one to derive the Gegenbauerpolynomial representation (54). To the best of my knowledge,this had been achieved by Niukkanen [63] in 1984, who in-troduced a fairly large class of exponentially decaying func-tions [63, Eqs. (2) and (3)], which contain all function setsconsidered in this article as special cases. The radial partof the Fourier transform of Niukkanen’s function can be ex-pressed in terms of an Appell function F [63, Eq. (21)],which is an hypergeometric function in two variables [25, Eq. (16.13.2)]. By means of a reduction formula in combinationwith a suitable quadratic transformation of a F , Niukkanen[63, Eq. (55)] obtained the Gegenbauer polynomial represen-tation (54). This Gegenbauer representation had also been de-rived by Belki´c and Taylor [64, Eq. (21)] in 1989 in connec-tion with their restricted version of Eq. (36) [64, Eq. (15)].Y¨ukc¸ ¨u and Y¨ukc¸ ¨u [2] used either a representation givenby their Eqs. (14) and (16) involving a non-terminating F ,which correspond to Eq. (35), or alternatively a Gegenbauerpolynomial representation given by their Eqs. (13) and (15),which correspond to Eq. (54). The non-terminating F inEq. (35) converges only for | p /α | < , whereas the Gegen-bauer polynomial in Eq. (54) is meaningful for all | p | ∈ R .Thus, Y¨ukc¸ ¨u and Y¨ukc¸ ¨u had to prove that their Gegenbauerpolynomial representation provides an analytic continuationof their representation involving a non-terminating F witha finite radius of convergence to all | p | ∈ R . They did thisby showing in [2, Table 1] that the radial parts of these repre-sentation give for a variety of quantum numbers n and ℓ andfor certain values of p identical numerical results. This highlypedestrian approach is no substitute for a rigorous mathemati-cal proof.So far, I only showed that representations involving a F with real argument can be obtained from representations in-volving a F with complex argument (compare Eqs. (37)and (43)). However, the inverse operations are also possible.For example, the application of the quadratic transformation[51, p. 51] F ( a, b ; 2 b ; z ) = (1 − z/ − a × F (cid:18) a , a + 12 ; b + 12 ; z [2 − z ] (cid:19) (55)to the F s in Eqs. (37) and (43) yields Eqs. (35) and (42).By suitably combining linear and quadratic transformations,many explicit expressions for the Fourier transform of a Slater-type function can be derived. However, this is not yet theend of the story. Those Gaussian hypergeometric functions F , for which a quadratic transformation exists, can also beexpressed in terms of Legendre functions [51, pp. 51 - 54].Since, however, Legendre functions can be viewed to be noth-ing but special hypergeometric series F [25, § F is ex-tremely useful in this context. It allows the derivation of alarge variety of different representations, which are all ana-lytic continuations of the basic expressions (35) and (37).The derivation and classification of the various expressionsfor the Fourier transforms of Slater-type functions is certainlyan achievement in its own right. Nevertheless, one should notforget that in the context of the Fourier transform of a bound-state hydrogen eigenfunction or of other functions based onthe generalized Laguerre polynomials, these Slater results areessentially irrelevant. The formulas presented in this Sectionconfirm once more what I had already emphasized in [23, p.29]: although extremely simple in the coordinate representa-tion, Slater-type functions are comparatively complicated ob-ject in momentum space. Their Fourier transforms have thesame level of complexity as the Fourier transforms of boundstate hydrogen eigenfunctions (see [5, Section IV]).Therefore, it cannot be a good idea to express the Fouriertransform of a bound-state hydrogen eigenfunction as a lin-ear combination of Fourier transforms of Slater-type functions.Because of strictly alternating sings, Eq. (33) as well as all for-mulas derived from it become numerically unstable for largequantum numbers n . In addition, these linear combinations ofthe Fourier transforms of Slater-type functions are for large n hopelessly inefficient compared to the classic result (29) de-rived by Podolsky and Pauling [1, Eq. (28)]. To the best of myknowledge, nobody has ever been able to construct Eq. (29)from a linear combination of Fourier transforms of Slater-typefunctions.If we evaluate the Fourier transform of a bound-state hy-drogen eigenfunction or of related functions via linear combi-nations of the Fourier transforms of Slater-type functions, wehave to deal with extensive intrinsic cancellations. I learnedthe hard way from my work on convergence acceleration andthe summation of divergent series (see for example [73, 74] or[25, § VI. EXPANSION IN TERMS OF REDUCED BESSELFUNCTIONS
A singe power z n is obviously simpler than a generalizedLaguerre polynomial L ( α ) n ( z ) . Therefore, it is tempting to be-lieve that powers produce simpler Hankel-type integrals thancorresponding generalized Laguerre polynomials. However,simplicity is a very elusive concept, and the results in Sec-tion V show that this seemingly obvious assumption is nottrue.If we want to evaluate the Fourier transforms of bound-statehydrogen eigenfunctions or of related functions by expandingthe generalized Laguerre polynomials, we must find alterna-tive expansion functions that have more convenient propertiesthan powers. The so-called reduced Bessel functions and theiran-isotropic generalization, the so-called B functions producethe desired expansions. Based on previous work by Shavitt[75, Eq. (55) on p. 15], B functions were defined in 1978 byFilter and Steinborn [76, Eq. (2.14)] as follows: B mn,ℓ ( β, r ) = ˆ k n − / ( βr )2 n + ℓ ( n + ℓ )! Y mℓ ( β r ) . (56)Here, β > , n ∈ Z , and ˆ k n − / is a reduced Bessel function.If K ν ( z ) is a modified Bessel function of the second kind [25,Eq. (10.27.4)], the reduced Bessel function is defined as fol-lows [77, Eqs. (3.1) and (3.2)]: ˆ k ν ( z ) = (2 /π ) / z ν K ν ( z ) , ν, z ∈ C . (57)If the order ν is half-integral, ν = n + 1 / with n ∈ N , thereduced Bessel function can be expressed as an exponentialmultiplied by a terminating confluent hypergeometric series F (see for example [78, Eq. (3.7)]): ˆ k n +1 / ( z ) = 2 n (1 / n e − z F ( − n ; − n ; 2 z ) . (58)A condensed review of the history of B functions includingnumerous references can be found in [79]. Reduced Besseland B functions had been the topic of my Diploma [80] andmy PhD thesis [81].Equations (56) – (58) indicate that B functions are fairlycomplicated mathematical objects. Therefore, it is not at allobvious why B functions should offer any advantages. How-ever, the Hankel-type integral [52, Eq. (2) on p. 410]) Z ∞ K µ ( αt ) J ν ( βt ) t µ + ν +1 d t = Γ( µ + ν + 1) 2 µ + ν α µ β ν [ α + β ] µ + ν +1 , ℜ ( µ + ν ) > |ℜ ( µ ) | , ℜ ( α ) > |ℜ ( β ) | , (59)implies that a B function possesses a Fourier transform of ex-ceptional simplicity: B mn,ℓ ( β, p ) = (2 π ) − / Z e − i p · r B mn,ℓ ( β, r ) d r = (2 /π ) / β n + ℓ − [ β + p ] n + ℓ +1 Y mℓ ( − i p ) . (60)0This is the most consequential and also the most often citedresult of my PhD thesis [81, Eq. (7.1-6) on p. 160]. Later,the Fourier transform (60) was published in [20, Eq. (3.7)].Independently and almost simultaneously, Eq. (60) was alsoderived by Niukkanen [63, Eqs. (57) - (58)].It follows from Eq. (58) that a B function can be expressedas a finite sum of Slater-type functions, or equivalently, thatthe Fourier transform (60) of a B function can be expressed asa linear combination of the Fourier transforms of Slater-typefunctions, just as Y¨ukc¸ ¨u and Y¨ukc¸ ¨u [2] had done it in the caseof bound-state hydrogen eigenfunctions (compare Section V).Y¨ukc¸ ¨u [71] used this seemingly simple approach of express-ing a B function as a linear combination of Slater-type func-tions [71, Eq. (21)]. For the Fourier transform of a Slater-typefunction – his Eqs. (32), (39), and (40) – he used the same ex-pressions as the ones used by Y¨ukc¸ ¨u and Y¨ukc¸ ¨u [2, Eqs. (13)- (16)]. This leads to explicit expressions [71, Eqs. (41) and(42)] that are, however, much more complicated and thereforemuch less useful than the remarkably compact Fourier trans-form (60).We do not know for sure whether Y¨ukc¸ ¨u and Y¨ukc¸ ¨u [2]were aware of the Fourier transform (29) derived by Podol-sky and Pauling [1, Eq. (28)] or of the other earlier referencesmentioned in Section I. Maybe, Y¨ukc¸ ¨u and Y¨ukc¸ ¨u genuinelybelieved that their results for the Fourier transform of a bound-state hydrogen eigenfunctions are actually the best possible.However, Y¨ukc¸ ¨u [71] did not only present his fairly compli-cated Eqs. (41) and (42) for the Fourier transform of a B func-tion, but as his Eq. (28) also the very compact expression (60).It is hard to imagine that anyone would want to use Y¨ukc¸ ¨u’scomplicated Eqs. (41) and (42) instead of the much simpler Eq. (60). Not all expressions, which are mathematically cor-rect, are useful and deserve to be published.The exceptionally simple Fourier transform (60) gives B functions a special position among exponentially decayingfunctions. It explains why other exponentially decaying func-tions as for example Slater-type functions with integral prin-cipal quantum numbers, bound state hydrogen eigenfunctions,and other functions based on generalized Laguerre polynomi-als can be expressed in terms of finite linear combinations of B functions (for details, see [5, Section IV] or [82, Section4]).The Fourier transform (60) was extensively used by Safouhiand co-workers for the evaluation of molecular multicenter in-tegrals with the help of numerical quadrature combined withextrapolation techniques. Many references of the Safouhigroup can be found in the PhD thesis of Slevinsky [83].Apart from the Fourier transform (60), the most importantexpression of this Section is the expansion of a generalized La-guerre polynomial in terms of reduced Bessel functions withhalf-integral indices [81, Eq. (3.3-35) on p. 45]: e − z L ( α ) n (2 z ) = (2 n + α + 1) × n X ν =0 ( − ν Γ( n + α + ν + 1) ν !( n − ν )!Γ( α + 2 ν + 2) ˆ k ν +1 / ( z ) . (61)This relationship was used by Filter and Steinborn [50, Eq.(3.17)] for the construction of addition theorems and other ex-pansions in terms of Lambda functions.With the help of Eq. (61), it is trivially simple to expressSturmians and Lambda functions as finite linear combinationsof B functions [5, Eqs. (4.19) and (4.20)]: Ψ mn,ℓ ( β, r ) = (2 β ) / ℓ (2 ℓ + 1)!! (cid:20) n ( n + ℓ )!( n − ℓ − (cid:21) / n − ℓ − X ν =0 ( − n + ℓ + 1) ν ( n + ℓ + 1) ν ν ! ( ℓ + 3 / ν B mν +1 ,ℓ ( β, r ) , (62) Λ mn,ℓ ( β, r ) = (2 β ) / ℓ (2 n + 1)(2 ℓ + 3)!! (cid:20) ( n + ℓ + 1)!( n − ℓ − (cid:21) / n − ℓ − X ν =0 ( − n + ℓ + 1) ν ( n + ℓ + 2) ν ν ! ( ℓ + 5 / ν B mν +1 ,ℓ ( β, r ) . (63)Now, we only need the Fourier transform (60) of a B function to obtain explicit expressions for the Fourier transforms of a Stur-mian or of a Lambda function. By combining Eqs. (60), (62) and (63), we obtain the following hypergeometric representations: Ψ mn,ℓ ( β, p ) = (2 π ) − / Z e − i p · r Ψ mn,ℓ ( β, r ) d r = 1(2 ℓ + 1)!! (cid:20) βπ n ( n + ℓ )!( n − ℓ − (cid:21) / (cid:20) ββ + p (cid:21) ℓ +2 Y mℓ ( − i p ) F (cid:18) − n + ℓ + 1 , n + ℓ + 1; ℓ + 32 ; β β + p (cid:19) , (64) Λ mn,ℓ ( β, p ) = (2 π ) − / Z e − i p · r Λ mn,ℓ ( β, r ) d r = (2 n + 1)(2 ℓ + 3)!! (cid:20) β ( n + ℓ + 1)! π ( n − ℓ − (cid:21) / (cid:20) ββ + p (cid:21) ℓ +2 Y mℓ ( − i p ) F (cid:18) − n + ℓ + 1 , n + ℓ + 2; ℓ + 52 ; β β + p (cid:19) . (65)The terminating F in Eq. (64) can according to Eq. (53) be replaced as a Gegenbauer polynomial, yielding Eq. (28) [5, Eq.(4.24)], and the terminating F in Eq. (65) can be expressed as a Jacobi polynomial [51, p. 212] via P ( α,β ) n ( x ) = (cid:18) n + αn (cid:19) F (cid:18) − n, α + β + n + 1; α + 1; 1 − x (cid:19) , (66)1yielding the following explicit expressions for the Fourier transforms of a Lambda function [5, Eq. (4.25)]: Λ mn,ℓ ( β, p ) = 2(1 / n (cid:26) β ( n + ℓ + 1)! ( n − ℓ − π (cid:27) / (cid:20) ββ + p (cid:21) ℓ +2 Y mℓ ( − i p ) P ( ℓ +3 / ,ℓ +1 / n − ℓ − (cid:18) p − β p + β (cid:19) . (67)The orthogonality relationships satisfied by the Fouriertransforms of Sturmians and Lambda functions with respectto an integration over the whole three-dimensional momentumspace can be deduced directly from the known orthogonalityproperties of the Gegenbauer and Jacobi polynomials [5, Eqs.(4.31) - (4.37)].My approach, which is based on the Eqs. (60) and (61), canalso be employed in the case of other, more complicated expo-nentially decaying functions. In [84, Abstract or Eqs. (1) and(2)], Guseinov introduced a large class of complete and or-thonormal functions. In terms of the polynomials (cid:2) L pq ( x ) (cid:3) BS defined in Eq. (3), Guseinov’s functions can be expressed asfollows: Ψ αnℓm ( ζ, r ) = ( − α (cid:20) (2 ζ ) ( n − ℓ − n ) α ( n + ℓ + 1 − α )! (cid:21) / × (2 ζr ) ℓ e − ζr (cid:2) L ℓ +2 − αn + ℓ +1 − α (cid:3) BS (2 ζr ) S ℓm ( θ, ϕ ) . (68)Here, ζ > is a scaling parameter, and S ℓm ( θ, ϕ ) is either areal or a complex spherical harmonic (Guseinov did not pro-vide an exact definition of S ℓm ( θ, ϕ ) ).The additional parameter α , which Guseinov calls fric-tional or self-frictional quantum number , was originally cho-sen to be an integer satisfying α = 1 , , − , − , · · · [84, Ab-stract]. In the text following [84, Eq. (3)], Guseinov remarkedthat for fixed α = 1 , , − , − , · · · the functions (68) form a complete orthonormal set .This statement is meaningless. Completeness is not a gener-ally valid property of a given function set. It only guaranteesthat functions belonging to a suitable Hilbert space, which hasto be specified, can be expanded by this function set, and thatthe resulting expansions converge with respect to the norm ofthis Hilbert space (for further details, I recommend a book byHiggins [38] or a review by Klahn [33]).Guseinov’s original definition (68) implies that his func-tions are according to [84, Eq. (4)] orthogonal with respectto the weight function w ( r ) = [ n ′ / ( ζr )] α [84, Eq. (4)]: Z [Ψ αnℓm ( ζ, r )] ∗ (cid:18) n ′ ζr (cid:19) α Ψ αn ′ ℓ ′ m ′ ( ζ, r ) d r = δ nn ′ δ ℓℓ ′ δ mm ′ . (69)In the theory of classical orthogonal polynomials, which is in-timately linked to Hilbert space theory, it is common practiceto introduce on the basis of their orthogonality relationshipssuitable inner products ( f | g ) w = R ba w ( x )[ f ( x )] ∗ g ( x )d x witha positive weight function w : [ a, b ] → R + . These weightedinner products then lead to the corresponding weightedHilbert spaces H w in which the orthogonal polynomials un-der consideration are complete and orthogonal. In the case of Guseinov’s orthogonality relationship (69),this approach does not work. The weight function w ( r ) =[ n ′ / ( ζr )] α cannot be used to define a Hilbert space becauseboth ζ and n ′ are in general undefined. Thus, instead of in-corporating ζ and n ′ into the weight function, they should beincorporated in the normalization factor.A further disadvantage of Guseinov’s original definition(68) is its use of the polynomials (cid:2) L pq ( x ) (cid:3) BS defined by Eq. (3),which can only have integral superscripts. As an alternative,I suggested the following definition, which uses the modernmathematical notation for the generalized Laguerre polynomi-als (see for example [85, Eq. (4.16)] or [23, Eq. (2.13)]): k Ψ mn,ℓ ( β, r ) = (cid:20) (2 β ) k +3 ( n − ℓ − n + ℓ + k + 2) (cid:21) / × e − βr L (2 ℓ + k +2) n − ℓ − (2 βr ) Y mℓ (2 β r ) . (70)The indices satisfy n ∈ N , ℓ ∈ N ≤ n − , − ℓ ≤ m ≤ ℓ , andthe scaling parameter satisfies β > .In my original definition in [85, Eq. (4.16)] or [23, Eq.(2.13)], I had assumed that k is a positive or negative inte-ger satisfying k = − , , , , . . . , which corresponds to thestraightforward translation − α k of Guseinov’s originalcondition α = 1 , , − , − , . . . [84, Eq. (4)]. Therefore, myoriginal definition in [85, Eq. (4.16)] or [23, Eq. (2.13)] as-sumed k being integral and contained ( n + ℓ + k + 1)! insteadof Γ( n + ℓ + k + 2) .However, in the text following [85, Eq. (4.16) on p. 11] orin the text following [23, Eq. (2.13) on p. 27], I had empha-sized that the condition k = − , , , , . . . is unnecessarilyrestrictive and that it can be generalized to k ∈ [ − , ∞ ) . Mycriticism of Guseinov’s original definition (68) was implicitlyconfirmed by Guseinov himself. In his later articles [86, 87],Guseinov generalized his so-called frictional quantum numberfrom originally α = 1 , , − , − , · · · to α ∈ ( −∞ , , whichcorresponds to k ∈ ( − , ∞ ) in my notation. This changecould not be done with Guseinov’s original definition (68).Therefore, Guseinov finally had to use the modern mathe-matical notation for his functions (compare [86, Eqs. (1) - (5)]or [87, Abstract]). To disguise the obvious, Guseinov used inthese formulas instead of a generalized Laguerre polynomial aterminating confluent hypergeometric series F . Because ofEq. (2), Guseinov’s formulas are equivalent to my definition(70). Characteristically, Guseinov did not acknowledge mycontributions [85, Eq. (4.16)] or [23, Eq. (2.13)] to his func-tions. Because of [88], Guseinov cannot claim to be unawareof [85].For fixed k ∈ ( − , ∞ ) , Guseinov’s functions defined by2Eq. (70) satisfy the orthonormality relationship Z (cid:2) k Ψ mn,ℓ ( β, r ) (cid:3) ∗ r k k Ψ m ′ n ′ ,ℓ ′ ( β, r ) d r = δ nn ′ δ ℓℓ ′ δ mm ′ , (71)which implies that they are complete and orthonormal in theweighted Hilbert space L r k ( R )= n f : R → C (cid:12)(cid:12)(cid:12) Z r k | f ( r ) | d r < ∞ o . (72)For k = 0 , the functions k Ψ mn,ℓ ( β, r ) are identical to theLambda functions defined by Eq. (13). Thus, they are com-plete and orthonormal in the Hilbert space L ( R ) of squareintegrable functions defined by Eq. (7). For k = − , Guseinov’s functions yield apart from a differ-ent normalization the Sturmians (11), which are complete andorthonormal in the Sobolev space W (1)2 ( R ) , or complete andorthogonal in the weighted Hilbert space L /r ( R ) .Personally, I prefer Sturmians satisfying Eq. (11): W (1)2 ( R ) is a proper subspace of L ( R ) with some addi-tional advantageous features [33], whereas L /r ( R ) is not.For k = 0 , the weighted Hilbert spaces L r k ( R ) are gen-uinely different from the Hilbert space L ( R ) of square inte-grable functions. We neither have L ( R ) ⊂ L r k ( R ) nor L r k ( R ) ⊂ L ( R ) . In quantum physics, it is tacitly as-sumed that bound-state wave functions are square integrable[89]. However, approximation processes converging with re-spect to the norm of the weighted Hilbert space L r k ( R ) with k = − , could produce functions that are not square inte-grable. Obviously, this would lead to some embarrassing con-ceptual and technical problems.With the help of Eq. (61), Guseinov’s functions can be expressed as a finite sum of B functions [23, Eq. (2.22)]: k Ψ mn,ℓ ( β, r ) = (cid:26) β k +3 ( n + ℓ + k + 1)!2 k +1 ( n − ℓ − (cid:27) / (2 n + k + 1) Γ(1 /
2) ( ℓ + 1)!Γ (cid:0) ℓ + 2 + k/ (cid:1) Γ (cid:0) ℓ + [ k + 5] / (cid:1) × n − ℓ − X ν =0 ( − n + ℓ + 1) ν ( n + ℓ + k + 2) ν ( ℓ + 2) ν ν ! (cid:0) ℓ + 2 + k/ (cid:1) ν (cid:0) ℓ + [ k + 5] / (cid:1) ν B mν +1 ,ℓ ( β, r ) . (73)Now, we only need the Fourier transform (60) to obtain an explicit expression for the Fourier transform of Guseinov’s function: k Ψ mn,ℓ ( β, p ) = (2 π ) − / Z e − i p · r k Ψ mn,ℓ ( β, r ) d r = (cid:26) β k +1 ( n + ℓ + k + 1)! π k ( n − ℓ − (cid:27) / (2 n + k + 1) Γ(1 /
2) ( ℓ + 1)!2 ℓ +2 Γ (cid:0) ℓ + 2 + k/ (cid:1) Γ (cid:0) ℓ + [ k + 5] / (cid:1) Y mℓ ( − i p ) × (cid:18) ββ + p (cid:19) ℓ +2 3 F (cid:18) − n + ℓ + 1 , n + ℓ + 2 , ℓ + 2; ℓ + 2 + k , ℓ + k + 52 ; β β + p (cid:19) . (74)In this Fourier transform, the radial part is essentially a terminating generalized hypergeometric series F , which simplifies foreither k = − or k = 0 to yield the terminating Gaussian hypergeometric series F in the hypergeometric representations (64)or (65) for the Fourier transforms of Sturmians and Lambda functions, respectively. Thus, the Fourier transform of Guseinov’sfunction k Ψ mn,ℓ ( β, r ) with k = − , is more complicated than the Fourier transforms of either Sturmians or Lambda functions. VII. SUMMARY AND CONCLUSIONS
The Fourier transform (29) for a bound-state hydrogeneigenfunction (5) is a classic result of quantum physics al-ready derived in 1929 by Podolsky and Pauling [1, Eq. (28)]with the help of the generating function (17). I am not awareof a more compact and more useful expression for this Fouriertransform.Y¨ukc¸ ¨u and Y¨ukc¸ ¨u [2], who were apparently unaware notonly of Podolsky and Pauling, but also of the other referenceslisted in Section I, proceeded differently. As discussed in Sec-tion V, Y¨ukc¸ ¨u and Y¨ukc¸ ¨u [2] expressed a generalized La-guerre polynomial as a finite sum of powers according toEq. (30), or equivalently, they expressed a bound-state hy- drogen eigenfunction as a finite sum of Slater-type functions.Since Fourier transformation is a linear operation, this leads toan expression of the Fourier transformation of a bound-statehydrogen eigenfunction as a finite sum of Fourier transformsof Slater-type functions, for which many explicit expressionsare known in the literature (compare Section V).At first sight, this approach, which requires no mathemat-ical skills, looks like a good idea. Unfortunately, the sim-plicity of Slater-type functions in the coordinate representa-tion is deceptive. As already emphasized in [23, p. 29], theFourier transforms of bound-state hydrogen eigenfunctionsand Slater-type function have the same level of complexity.Consequently, it cannot be a good idea to express the Fouriertransform of a bound-state hydrogen eigenfunction as a lin-3ear combination of Fourier transforms of Slater-type functions.Moreover, in the case of large principal quantum numbers n ,these finite sums tend to become numerically unstable. Thisis a direct consequence of the alternating signs in Eq. (30).In principle, it should be possible to derive thePodolsky and Pauling formula (29) from the compar-atively complicated linear combinations presented byY¨ukc¸ ¨u and Y¨ukc¸ ¨u. However, the Fourier transforms ofSlater-type functions discussed in Section V are all fairlycomplicated objects. Therefore, it is very difficult or evenpractically impossible to obtain the remarkably compactPodolsky and Pauling formula (29) in this way. I am notaware of anybody who achieved this.It is nevertheless possible to derive thePodolsky and Pauling formula (29) by expanding gener-alized Laguerre polynomials, albeit in terms of some other,less well known polynomials. This was shown in Section VI.The key relationships in Section VI are the exceptionallysimple Fourier transform (60) of a B function and the expan-sion (61) of a generalized Laguerre polynomial in terms ofreduced Bessel functions (58) with half-integral indices. Withthe help of Eqs. (60) and (61) it is trivially simple to derivethe Podolsky and Pauling formula (29). This derivation ismuch simpler than the original derivation by Podolsky andPauling [1], which is discussed in Section IV and whichrequired the skillful use of the generating function (17) of thegeneralized Laguerre polynomials.Hylleraas [35] observed already in 1928 that bound-statehydrogen eigenfunctions without the inclusion of the mathe-matically very difficult continuum eigenfunctions are incom-plete in the Hilbert space of square integrable functions (com-pare Section III). This is a highly consequential fact, whichY¨ukc¸ ¨u and Y¨ukc¸ ¨u were apparently not aware of. In combina-tion with the difficult nature of the continuum eigenfunctions,this incompleteness greatly limits the practical usefulness ofbound-state eigenfunctions as mathematical tools. It is cer-tainly not a good idea to do expansions in terms of an incom- plete function set.Therefore, attention has shifted away from hydrogen eigen-functions to other, related function sets also based on the gen-eralized Laguerre polynomials, which, however, have moreconvenient completeness properties. The best known exam-ples are the so-called Sturmians (11), which had been intro-duced by Hylleraas [35] already in 1928 and which can beobtained from the bound-state hydrogen eigenfunctions by thesubstitution Z/n β according to Eq. (12), and the so-calledLambda functions (13), which were also introduced by Hyller-aas [46] in 1929.With the help of the expansion (61), it is a trivial matter toexpress both Sturmians and Lambda functions as linear combi-nations of B functions, yielding Eqs. (62) and (63). Then, oneonly needs the Fourier transform (60) to convert these linearcombinations to compact explicit expressions for the Fouriertransforms of Sturmians and Lambda functions, respectively.In 2002, Guseinov [84] introduced a large class of com-plete and orthonormal functions defined by Eq. (68), whichused an antiquated notation for the Laguerre polynomials. Gu-seinov’s functions contain an additional parameter α called frictional or self-frictional quantum number , which was origi-nally assumed to be integral. Depending on this α , Guseinov’sfunctions contain Sturmians and Lambda functions as specialcases.Guseinov’s original notation (68) does not allow non-integral values of α . In order to rectify this obvious deficiency,I introduced in [23, 85] the alternative definition (70) whichuses the modern mathematical notation for the generalizedLaguerre polynomials. 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