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January18,2013 3:17 Integral TransformsandSpecial Functions gegpoly Integral Transforms and Special Functions Vol. 00, No. 00, January 2009,1–12 RESEARCH ARTICLE On a generalization of the generating function for Gegenbauer polynomials 3 Howard S. Cohla∗ 1 0 aApplied and Computational Mathematics Division, National Institute of Standards and 2 Technology, Gaithersburg, Maryland, U.S.A. n a (v3.5 released October2008) J 7 Ageneralizationof thegenerating functionforGegenbauer polynomialsisintroducedwhose 1 coefficientsaregivenintermsofassociatedLegendrefunctionsofthesecondkind.Wediscuss howourexpansionrepresentsageneralizationofseveralpreviouslyderivedformulaesuchas ] Heine’s formula and Heine’s reciprocal square-root identity. We also show how this expan- A sioncanbeusedtocomputehypersphericalharmonicexpansionsforpower-lawfundamental C solutionsofthepolyharmonicequation. . h t Keywords:Euclideanspace;Polyharmonicequation; Fundamental solution;Gegenbauer a polynomials;associatedLegendrefunctions m AMS Subject Classification: 35A08;35J05;32Q45; 31C12;33C05;42A16 [ 3 v 5 1. Introduction 3 7 Gegenbauer polynomials Cν(x) are given as the coefficients of ρn for the generat- 2 n . ing function (1+ρ2 2ρx)−ν. The study of these polynomials was pioneered in a 5 − series of papers by Leopold Gegenbauer (Gegenbauer (1874,1877,1884,1888,1893) 0 1 [13–17]).Themain resultwhichthis paperrelies uponis Theorem2.1 below.This 1 theorem gives a generalized expansion over Gegenbauer polynomials Cµ(x) of the n v: algebraic function z (z x)−ν. Our proof is combinatoric in nature and has i 7→ − X great potential for proving new expansion formulae which generalize generating functions. Our methodology can in principle be applied to any generating function r a for hypergeometric orthogonal polynomials, of which there are many (see for instance Srivastava & Manocha (1984) [25]; Erdélyi et al. (1981) [11]). The concept of the proof is to start with a generating function and use a connection formula to expresstheorthogonal polynomialasafiniteseriesinpolynomialsofthesametype withdifferentparameters.Theresultingformulaewillthenproducenewexpansions for the polynomials which result from a limiting process, e.g., Legendre polynomials and Chebyshev polynomials of the first and second kind. Connection formulae for classical orthogonal polynomials and their q-extensions are well-known (see Is- mail (2005) [20]). In this paper we applied this method of proof to the generating function for Gegenbauer polynomials. Thispaperisorganizedasfollows.InSection2wederiveacomplexgeneralization ofthegeneratingfunctionforGegenbauer polynomials.InSection3wediscusshow ∗Correspondingauthor.Email:[email protected] ISSN:1065-2469print/ISSN1476-8291online (cid:13)c 2009Taylor&Francis DOI:10.1080/10652460YYxxxxxxx http://www.informaworld.com January18,2013 3:17 Integral TransformsandSpecial Functions gegpoly 2 Howard S. Cohl our complex generalization reduces to previously derived expressions and leads to extensions in appropriate limits. In Section 4 we use our complex expansion to generalize aformula originally developed by Sack (1964) [24] on R3,to computean expansion in terms of Gegenbauer polynomials for complex powers of the distance between two points on a d-dimensional Euclidean space for d 2. ≥ Throughoutthispaperwerelyonthefollowingdefinitions.Fora ,a ,a ,... C, 1 2 3 if i,j Z and j < i then j a = 0 and j a = 1, where C represents∈the ∈ n=i n n=i n complex numbers. The set of natural numbers is given by N := 1,2,3,... , the P Q { } set N := 0,1,2,... = N 0 , and the set Z := 0, 1, 2,... . The sets Q 0 { } ∪{ } { ± ± } and R represent the rational and real numbers respectively. 2. Generalization of the generating function for Gegenbauer polynomials We present the following generalization of the generating function for Gegenbauer polynomials whose coefficients are given in terms of associated Legendre functions of the second kind. Theorem 2.1 Let ν C N , µ ( 1/2, ) 0 and z C ( ,1] on any 0 ∈ \− ∈ − ∞ \{ } ∈ \ −∞ ellipse with foci at 1 with x in the interior of that ellipse. Then ± 1 = 2µ+1/2Γ(µ)eiπ(µ−ν+1/2) ∞ (n+µ)Qν−µ−1/2(z)Cµ(x). (1) (z x)ν √πΓ(ν)(z2 1)(ν−µ)/2−1/4 n+µ−1/2 n − − n=0 X If one substitutes z = (1+ρ2)/(2ρ) in (1) with 0 < ρ < 1, then one obtains an | | alternate expression with x [ 1,1], ∈ − 1 Γ(µ)eiπ(µ−ν+1/2) = (1+ρ2 2ρx)ν √πΓ(ν)ρµ+1/2(1 ρ2)ν−µ−1/2 − − ∞ (n+µ)Qν−µ−1/2 1+ρ2 Cµ(x). (2) × n+µ−1/2 2ρ n n=0 (cid:18) (cid:19) X One can see that by replacing ν = µ in (2), and using (8.6.11) in Abramowitz & Stegun(1972)[1],thattheseformulaearegeneralizations ofthegeneratingfunction for Gegenbauer polynomials (first occurence in Gegenbauer (1874) [13]) ∞ 1 = Cν(x)ρn, (3) (1+ρ2 2ρx)ν n − n=0 X where ρ C with ρ < 1 and ν ( 1/2, ) 0 (see for instance (18.12.4) in ∈ | | ∈ − ∞ \{ } Olver et al. (2010) [23]). The Gegenbauer polynomials Cν : C C can be defined n → by (2ν) 1 1 x Cν(x) := n F n,n+2ν;ν + ; − , (4) n n! 2 1 − 2 2 (cid:18) (cid:19) where n N , ν ( 1/2, ) 0 , and F : C2 (C N ) z C : 0 2 1 0 ∈ ∈ − ∞ \ { } × \ − × { ∈ z < 1 C, the Gauss hypergeometric function, can be defined in terms of the | | } → January18,2013 3:17 Integral TransformsandSpecial Functions gegpoly Integral Transforms and Special Functions 3 following infinite series ∞ (a) (b) zn n n F (a,b;c;z) := (5) 2 1 (c) n! n n=0 X (see (2.1.5) in Andrews, Askey & Roy 1999), and elsewhere by analytic continua- tion. The Pochhammer symbol (rising factorial) () : C C is defined by n · → n (z) := (z+i 1), n − i=1 Y where n N . For the Gegenbauer polynomials Cν(x), we refer to n and ν as the ∈ 0 n degree and order respectively. Proof Consider the generating function for Gegenbauer polynomials (3). The connection relation which expresses a Gegenbauer polynomial with order ν as a sum over Gegenbauer polynomials with order µ is given by Cν(x) = (2ν)n n (ν +k+ 12)n−k(2ν +n)k(µ+ 21)kΓ(2µ+k) n (ν + 1) (n k)!(2µ) Γ(2µ+2k) 2 n k=0 − k X n+k,n+k+2ν,µ+k+ 1 ×3F2 − ν +k+ 1,2µ+2k+1 2;1 Ckµ(x). (6) (cid:18) 2 (cid:19) This connection relation can be derived by starting with Theorem 9.1.1 in Ismail (2005) [20] combined with (see for instance (18.7.1) in Olver et al. (2010) [23]) (2ν) Cν(x) = n P(ν−1/2,ν−1/2)(x), (7) n (ν + 1) n 2 n i.e., the Gegenbauer polynomials are given as symmetric Jacobi polynomials. The (α,β) Jacobi polynomials P : C C can be defined as (see for instance (18.5.7) in n → Olver et al. (2010) [23]) (α+1) 1 x P(α,β)(x) := n F n,n+α+β +1;α+1; − , (8) n n! 2 1 − 2 (cid:18) (cid:19) where n N , and α,β > 1 (see Table 18.3.1 in Olver et al. (2010) [23]). The 0 ∈ − generalized hypergeometricfunction F : C3 (C N )2 z C : z < 1 C 3 2 0 × \− ×{ ∈ | | } → can be defined in terms of the following infinite series ∞ a ,a ,a (a ) (a ) (a ) zn F 1 2 3;z := 1 n 2 n 3 n . 3 2(cid:18) b1,b2 (cid:19) n=0 (b1)n(b2)n n! X If we replace the Gegenbauer polynomial in the generating function (3) using the connection relation (6), we obtain a double summation expression over k and n. By reversing the order of the summations (justification by Tannery’s theorem) and shifting the n-index by k, we obtain after making some reductions and simplifica- January18,2013 3:17 Integral TransformsandSpecial Functions gegpoly 4 Howard S. Cohl tions, the following double-summation representation ∞ 1 √πΓ(µ) ρk µ+k µ = C (x) (1+ρ2 2ρx)ν 22ν−1Γ(ν) k 22k Γ(ν +k+ 1)Γ(µ+k+1) − k=0 2 X ∞ Γ(2ν +2k) n,n+2k+2ν,µ+k+ 1 × n! 3F2 −ν +k+ 1,2µ+2k+1 2;1 . (9) n=0 (cid:18) 2 (cid:19) X The F generalized hypergeometric function appearing the above formula may be 3 2 simplified using Watson’s sum a,b,c F ;1 3 2 1(a+b+1),2c (cid:18)2 (cid:19) √πΓ c+ 1 Γ 1(a+b+1) Γ c+ 1(1 a b) = 2 2 2 − − , Γ 1(a+1) Γ 1(b+1) Γ c+ 1(1 a) Γ c+ 1(1 b) 2 (cid:0) (cid:1)2 (cid:0) 2(cid:1) (cid:0)− 2 −(cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) where Re(2c a b) > 1 (see for instance (16.4.6) in Olver et al. [23]), therefore − − − 1 n,n+2k+2ν,µ+k+ 1 Γ(ν +k+ 1)Γ(µ+k+1) 3F2 −ν +k+ 1,2µ+2k+1 2;1 2 (cid:18) 2 (cid:19) √πΓ(µ ν +1) = − , (10) Γ 1−n Γ ν +k+ n+1 Γ µ+k+1+ n Γ µ ν +1 n 2 2 2 − − 2 (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) for Re(µ ν)> 1. By inserting (10) in (9), it follows that − − ∞ 1 πΓ(µ)Γ(µ ν +1) ρk µ = − (µ+k)C (x) (1+ρ2 2ρx)ν 22ν−1Γ(ν) k 22k − k=0 X ∞ ρnΓ(2ν +2k+n) . × n!Γ 1−n Γ ν +k+ n+1 Γ µ+k+1+ n Γ µ ν +1 n n=0 2 2 2 − − 2 X (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) It is straightforward to show using (5) and Γ(z) Γ(z n)= ( 1)n , − − ( z+1) n − for n N and z C N , and the duplication formula (i.e., (5.5.5) in Olver et 0 0 ∈ ∈ \− al. (2010) [23]) 22z−1 1 Γ(2z) = Γ(z)Γ z+ , √π 2 (cid:18) (cid:19) January18,2013 3:17 Integral TransformsandSpecial Functions gegpoly Integral Transforms and Special Functions 5 provided 2z N , that 0 6∈ − ∞ ρnΓ(2ν +2k+n) n!Γ 1−n Γ ν +k+ n+1 Γ µ+k+1+ n Γ µ ν +1 n n=0 2 2 2 − − 2 X (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) 22ν+2k−1Γ(ν +k) = F ν +k,ν µ;µ+k+1;ρ2 , 2 1 πΓ(µ+k+1)Γ(µ ν +1) − − (cid:0) (cid:1) so therefore ∞ 1 Γ(µ) = (µ+k)Cµ(x)ρk (1+ρ2 2ρx)ν Γ(ν) k − k=0 X Γ(ν +k) F ν +k,ν µ;µ+k+1;ρ2 . (11) 2 1 ×Γ(µ+k+1) − (cid:0) (cid:1) Finally utilizing the quadratic transformation of the hypergeometric function a a+1 4z F (a,b;a b+1;z) = (1+z)−a F , ;a b+1; , 2 1 − 2 1 2 2 − (z+1)2 (cid:18) (cid:19) for z < 1 (see (3.1.9) in Andrews, Askey & Roy (1999) [2]), combined with the defin|i|tionoftheassociatedLegendrefunctionofthesecondkindQµ :C ( ,1] ν \ −∞ → C in terms of the Gauss hypergeometric function √πeiπµΓ(ν +µ+1)(z2 1)µ/2 Qµ(z) := − ν 2ν+1Γ(ν + 3)zν+µ+1 2 ν +µ+2 ν +µ+1 3 1 F , ;ν + ; , (12) ×2 1 2 2 2 z2 (cid:18) (cid:19) for z > 1 and ν + µ + 1 / N (cf. Section 14.21 and (14.3.7) in Olver et al. 0 | | ∈ − (2010) [23]), one can show that F ν +k,ν µ;µ+k+1;ρ2 2 1 − (cid:0) Γ(µ+k+(cid:1)1)eiπ(µ−ν+1/2) ν−µ−1/2 1+ρ2 = Q , √πΓ(ν +k)ρµ+k+1/2(1 ρ2)ν−µ−1/2 k+µ−1/2 2ρ − (cid:18) (cid:19) which when used in (11) produces (2). Since the Gegenbauer polynomial is just a symmetric Jacobi polynomial (7), through Theorem 9.1.1 in Szego˝ (1959) [26] (Expansion of an analytic function in a Jacobi series), since f : C C defined by z f (x) := (z x)−ν is analytic in [ 1,1], then the above expansion→in Gegenbauer z − − polynomials is convergent if the point z C lies on any ellipse with foci at 1 and x can lie on any point interior to that∈ellipse. ±(cid:4) January18,2013 3:17 Integral TransformsandSpecial Functions gegpoly 6 Howard S. Cohl 3. Generalizations, Extensions and Applications By considering in (2) the substitution ν = d/2 1 and the map ν ν/2, one − 7→ − obtains the formula ρ(d−1)/2(1 ρ2)ν−(d−1)/2 e−iπ(ν−(d−1)/2)Γ(d−2) − = 2 (1+ρ2 2ρx)ν 2√πΓ(ν) − ∞ (2n+d 2)Qν−(d−1)/2 1+ρ2 Cd/2−1(x). × − n+(d−3)/2 2ρ n n=0 (cid:18) (cid:19) X This formula generalizes (9.9.2) in Andrews, Askey & Roy (1999) [2]. Bytakingthelimitasµ 1/2in(1),oneobtainsageneralresultisanexpansion → over Legendre polynomials, namely 1 eiπ(1−ν)(z2 1)(1−ν)/2 ∞ = − (2n+1)Qν−1(z)P (x), (13) (z x)ν Γ(ν) n n − n=0 X using P (x) = C1/2(x), (14) n n which is clear by comparing (cf. (18.7.9) of Olver et al. (2010) and (4) or (8)) 1 x P (x) := F n,n+1;1; − , (15) n 2 1 − 2 (cid:18) (cid:19) and (4). If one takes ν = 1 in (13) then one has an expansion of the Cauchy denominator which generalizes Heine’s formula (see for instance Olver (1997) [22, Ex. 13.1]; Heine (1878) [18, p. 78]) ∞ 1 = (2n+1)Q (z)P (x). n n z x − n=0 X By taking the limit as µ 1 in (1), one obtains a general result which is an → expansion over Chebyshev polynomials of the second kind, namely 1 23/2eiπ(3/2−ν) ∞ ν−3/2 = (n+1)Q (z)U (x), (16) (z x)ν √πΓ(ν)(z2 1)ν/2−3/4 n+1/2 n − − n=0 X using (18.7.4) in Olver et al. (2010) [23], U (x) = C1(x). If one considers the case n n ν = 1 in (16) then the associated Legendre function of the second kind reduces to an elementary function through (8.6.11) in Abramowitz & Stegun (1972) [1], namely ∞ 1 U (x) n = 2 . z x (z+√z2 1)n+1 − nX=0 − By taking the limit as ν 1 in (1), one produces the Gegenbauer expansion of → the Cauchy denominator given in Durand, Fishbane & Simmons (1976) [10, (7.2)], January18,2013 3:17 Integral TransformsandSpecial Functions gegpoly Integral Transforms and Special Functions 7 namely ∞ 1 = 2µ+1/2Γ(µ)eiπ(µ−1/2)(z2 1)µ/2−1/4 (n+µ)Q−µ+1/2 (z)Cµ(x). z x √π − n+µ−1/2 n − n=0 X Using(2.4)therein,theassociatedLegendrefunctionofthesecondkindisconverted to the Gegenbauer function of the second kind. If one considers the limit as µ 0 in (1) using → n+µ lim Cµ(x) = ǫ T (x) µ→0 µ n n n (seeforinstance(6.4.13)inAndrews,Askey&Roy(1999)[2]),whereT :C Cis n → the Chebyshevpolynomial of thefirstkinddefinedas (see Section 5.7.2 in Magnus, Oberhettinger & Soni (1966) [21]) 1 1 x T (x) := F n,n; ; − , n 2 1 − 2 2 (cid:18) (cid:19) and ǫ = 2 δ is the Neumann factor, commonly appearing in Fourier cosine n n,0 − series, then one obtains 1 2e−iπ(ν−1/2)(z2 1)−ν/2+1/4 ∞ ν−1/2 = − ǫ T (x)Q (z). (17) (z x)ν π Γ(ν) n n n−1/2 − r n=0 X The result (17), which is a generalization of Heine’s reciprocal square-root identity (see Heine (1881) [19, p. 286]; Cohl & Tohline (1999) [9, (A5)]). Polynomials in (z x) also naturally arise by considering the limit ν n N . This limit is 0 − → ∈ − given in (4.4) of Cohl & Dominici (2010) [7], namely q 2 ( q) (z x)q = i( 1)q+1 (z2 1)q/2+1/4 ǫ T (x) − n Qq+1/2(z), (18) − − π − n n (q+n)! n−1/2 r n=0 X for q N . 0 ∈ Notethatalloftheaboveformulaearerestrictedbytheconvergencecriteriongiven by Theorem 9.1.1 in Szego˝ (1959) [26] (Expansion of an analytic function in a Jacobi series), i.e., since the functions on the left-hand side are analytic in [ 1,1], − then the expansion formulae are convergent if the point z C lies on any ellipse ∈ with fociat 1then x can lie on any point interior to that ellipse. Except of course ± (18) which converges for all points z,x C since the function on the left-hand side ∈ is entire. Aninterestingextensionoftheresultspresentedinthispaper,originallyuploaded toarXivinCohl(2011)[6]havebeenobtainedrecentlyinSzmytkowski(2011) [28], to obtain formulas such as ∞ n+µPν−µ (t)Cµ(x) = √π(1−t2)(ν−µ)/2 µ n+µ−1/2 n 2µ−1/2Γ(µ+1)Γ 1 ν n=0 2 − X (cid:0) (cid:1) 0 if 1 < x < t < 1, − ×((x t)−ν−1/2 if 1 < t < x < 1, − − January18,2013 3:17 Integral TransformsandSpecial Functions gegpoly 8 Howard S. Cohl and ∞ n+µQν−µ (t)Cµ(x) = √πΓ ν + 12 (1−t2)(ν−µ)/2 µ n+µ−1/2 n 2µ+1/2Γ(µ+1) n=0 (cid:0) (cid:1) X (t x)−ν−1/2 if 1 < x < t < 1, − − ×((x t)−ν−1/2cos[π(ν + 1)] if 1 < t < x < 1, − 2 − where Reµ > 1/2, Reν < 1/2 and Pµ,Qµ : ( 1,1) C are Ferrers functions ν ν − − → (associated Legendrefunctionson-the-cut)ofthefirstandsecondkind.TheFerrers functions of the first and second kind can be defined using Olver et al. (2010) [23, (14.3.11-12)]. 4. Expansion of a power-law fundamental solution of the polyharmonic equation A fundamental solution for the polyharmonic equation on Euclidean space Rd is a function d :(Rd Rd) (x,x) :x Rd R which satisfies the equation Gk × \{ ∈ } → ( ∆)k d(x,x′)= δ(x x′), (19) − Gk − where ∆ : Cp(Rd) Cp−2(Rd), p 2, is the Laplacian operator on Rd defined by → ≥ d ∂2 ∆ := , ∂x2 i=1 i X x = (x ,...,x ),x′ = (x′,...,x′) Rd, and δ is the Dirac delta function. Note 1 d 1 d ∈ thatweintroduceaminussignintotheequations wheretheLaplacianisused,such asin(19),tomaketheresultingoperatorpositive.ByEuclideanspaceRd,wemean the normed vector space given by the pair (Rd, ), where : Rd [0, ) is k·k k·k → ∞ the Euclidean norm on Rd defined by x := x2+ +x2, with inner product k k 1 ··· d (, ) : Rd Rd R defined as q · · × → d ′ ′ (x,x) := x x . (20) i i i=1 X Then Rd is a C∞ Riemannian manifold with Riemannian metric induced from the inner product (20). Set Sd−1 = x Rd : (x,x) = 1 , then Sd−1, the (d 1)- dimensionalunithypersphere,isareg{ula∈rsubmanifoldof}RdandaC∞Rieman−nian manifold with Riemannian metric induced from that on Rd. Theorem 4.1 Let d,k N. Define ∈ ( 1)k+d/2+1 x x′ 2k−d ′ − k − k log x x β (k 1)! (k d/2)! 22k−1πd/2 k − k− p,d  − − Gkd(x,x′) :=  (cid:0) if d ev(cid:1)en, k ≥ d/2, Γ(d/2 k) x x′ 2k−d − k − k otherwise, (k 1)! 22kπd/2   −     January18,2013 3:17 Integral TransformsandSpecial Functions gegpoly Integral Transforms and Special Functions 9 where p = k −d/2, βp,d ∈ Q is defined as βp,d := 21 Hp+Hd/2+p−1−Hd/2−1 , with H Q being the jth harmonic number j ∈ (cid:2) (cid:3) j 1 H := , j i i=1 X then d is a fundamental solution for ( ∆)k on Euclidean space Rd. Gk − Proof See Cohl (2010) [4] and Boyling (1996) [3]. Consider the following functions gd,ld : (Rd Rd) (x,x) : x Rd R k k × \ { ∈ } → defined for d odd and for d even with k d/2 1 as a power-law, namely ≤ − gd(x,x′):= x x′ 2k−d, (21) k k − k and for d even, k d/2, with logarithmic behavior as ≥ ld(x,x′):= x x′ 2p log x x′ β , k k − k k − k− p,d (cid:0) (cid:1) with p = k d/2. By Theorem 4.1 we see that the functions gd and ld equal real − k k non-zero constant multiples of d for appropriate parameters. Therefore by (19), Gk gd and ld are fundamental solutions of the polyharmonic equation for appropriate k k parameters. In this paper, we only consider functions with power-law behavior, although in future publications we will consider the logarithmic case (see Cohl (2012) [5] for the relevant Fourier expansions). Now we consider theset of hypersphericalcoordinate systems which parametrize points on Rd. The Euclidean distance between two points represented in these coordinate systems is given by x x′ = √2rr′[z cosγ]1/2, k − k − where the toroidal parameter z (1, ), (2.6) in Cohl et al. (2001) [8], is given by ∈ ∞ r2+r′2 z := , ′ 2rr and the separation angle γ [0,π] is given through ∈ ′ (x,x) cosγ = , (22) rr′ ′ ′ ′ wherer,r [0, ) aredefinedsuchthatr := x andr := x .We willusethese ∈ ∞ k k k k quantities toderive Gegenbauer expansionsof power-law fundamentalsolutions for powers of the Laplacian gd (21) represented in hyperspherical coordinates. k Corollary 4.2 For d 3,4,5,... , ν C, x,x′ Rd with r = x , r′ = x′ , ∈ { } ∈ ∈ k k k k January18,2013 3:17 Integral TransformsandSpecial Functions gegpoly 10 Howard S. Cohl ′ ′ and cosγ = (x,x)/(rr ), the following formula holds eiπ(ν+d−1)/2Γ d−2 r2 r2 (ν+d−1)/2 x x′ ν = 2 >− < k − k 2√πΓ −ν2(cid:0) (cid:1)(cid:0) (rr′)((cid:1)d−1)/2 ∞ (2n+(cid:0)d (cid:1)2)Q(1−ν−d)/2 r2+r′2 Cd/2−1(cosγ), (23) × − n+(d−3)/2 2rr′ n n=0 (cid:18) (cid:19) X where r := min r,r′ . ≶ max{ } Note that (23) is seen to be a generalization of Laplace’s expansion on R3 (see for instance Sack (1964) [24]) ∞ 1 rn = < P (cosγ), x x′ rn+1 n k − k n=0 > X which is demonstrated by utilizing (14) and simplifying the associated Legendre 1/2 function of the second kind in (23) through Q :C ( ,1] C defined such −1/2 \ −∞ → that π Q1/2 (z) =i (z2 1)−1/4 −1/2 2 − r (cf. (8.6.10) in Abramowitz & Stegun (1972) [1] and (12)). The addition theorem for hyperspherical harmonics, which generalizes n 4π P (cosγ) = Y (x)Y (x′), (24) n n,m n,m 2n+1 m=−n X b b where P (x) is the Legendre polynomial of degree n N , for d= 3, is given by n 0 ∈ Γ(d/2) YK(x)YK(x′) = (2n+d 2)Cd/2−1(cosγ), (25) n n 2πd/2(d 2) − n K − X b b where K stands for a set of (d 2)-quantum numbers identifying degenerate har- − monicsforagivenvalueofnandd,andγ istheseparationangle(22).Thefunctions YK : Sd−1 C are the normalized hyperspherical harmonics, and Y : S2 C arne the nor→malized spherical harmonics for d = 3. Note that x,x′ Snd,k−1, are→the vectors of unit length in the direction of x,x′ Rd respectively. Fo∈r a proof of the ∈ addition theorem for hyperspherical harmonics (25), see Wen & Avery (1985) [30] b b and for a relevant discussion, see Section 10.2.1 in Fano & Rau (1996) [12]. The correspondence between (24) and (25) arises from (4) and (15) namely (14). One can usethe addition theorem for hypersphericalharmonics to expand a fundamental solution of the polyharmonic equation on Rd. Through the use of the addition theorem for hyperspherical harmonics we see that the Gegenbauer poly- d/2−1 nomials C (cosγ) are hyperspherical harmonics when regarded as a function n of xonly (seeVilenkin (1968) [29]). Normalization of thehypersphericalharmonics is accomplished through the following integral b YK(x)YK(x)dΩ = 1, n n ZSd−1 b b