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Some double-series identities and associated generating-function relationships
Applied Mathematics Letters 19 (2006) 887–893 www.elsevier.com/locate/aml
Some double-series identities and associated generating-function relationships Kung-Yu Chen a , Shuoh-Jung Liu a , H.M. Srivastava b,∗ a Department of Mathematics, Tamkang University, Tamsui 25137, Taiwan, ROC b Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3P4, Canada
Received 8 July 2005; accepted 22 July 2005
Abstract The main object of the present work is to investigate several families of double-series identities and their applications to the Srivastava–Daoust hypergeometric function in two variables. A number of associated generating-function relationships, involving certain classes of hypergeometric polynomials, are also considered. c 2005 Elsevier Ltd. All rights reserved. Keywords: Series identities; Srivastava–Daoust multivariable hypergeometric function; Generating-function relationships; Hypergeometric functions and polynomials; Cubic transformations; Generating functions
1. Introduction and definitions As usual, we denote by p F q a generalized hypergeometric function with p numerator and q denominator parameters (see, for details, [7,10]). Also, throughout this work, (λ)ν is the Pochhammer symbol (or the shifted factorial, since (1)n = n! for n ∈ N0 ) defined (for λ, ν ∈ C and in terms of the familiar Gamma function) by Γ (λ + ν) 1 (ν = 0; λ ∈ C \ {0}) = (λ)ν := λ(λ + 1) · · · (λ + n − 1) (ν = n ∈ N; λ ∈ C), Γ (λ) where N0 := N ∪ {0} (N := {1, 2, 3, . . .}). For m ∈ N, we find it to be convenient to abbreviate the array of m parameters λ+m−1 λ λ+1 , ,..., m m m simply by (m; λ), the array being understood to be empty when m = 0. ∗ Corresponding author. Tel.: +1 250 721 7455; fax: +1 250 721 8962.
E-mail addresses: [email protected] (K.-Y. Chen), [email protected] (S.-J. Liu), [email protected] (H.M. Srivastava). c 2005 Elsevier Ltd. All rights reserved. 0893-9659/$ - see front matter doi:10.1016/j.aml.2005.07.013
888
K.-Y. Chen et al. / Applied Mathematics Letters 19 (2006) 887–893
Motivated essentially by several recent investigations dealing with (for example) series identities and their various applications (cf. [6,8]; see also [5] and the other references cited in each of these earlier works), we prove two doubleseries identities and apply these identities in order to deduce hypergeometric reduction formulas and generatingfunction relationships associated with the Srivastava–Daoust hypergeometric function in two variables, which is defined in Section 3 below. 2. Series identities based upon Bailey’s cubic transformations Over seven decades ago, the following two cubic transformations were proven by Wilfrid Norman Bailey (1893–1961) (cf. [3,4]; see also [2]): a, 2b − a − 1, a − 2b + 2; x (3; a); 27x 3 3 − (2.1) = (1 − x)−a 3 F 2 3F2 b, a − b + ; 4 b, a − b + ; 4(1 − x)3 2 2 and 1 (3; a); 27x 2 x −a a, b − , a − b + 1; 3 . (2.2) x = 1− 2 3F2 3 F2 b, a − b + ; (4 − x)3 4 2b − 1, 2a − 2b + 2; 2 Bailey’s cubic transformation (2.2) is recorded erroneously by Andrews et al. [1, p. 185, Exercise 38]. Making use of (2.1) and (2.2), we first prove Theorem 1 below. Theorem 1. Let {Ω (n)}∞ n=0 be a bounded sequence of complex numbers. Then 1 ∞ b − 2 n (2b − 1)2m+n x m y n Ω (m + n) (b)m (2b − 1)n m! n! m,n=0 b − 12 ∞ 2 m n 2m+n (x y ) (4x + y) (b, 2b − 1 ∈ Z− Ω (3m + n) = 0) (b) m! n! m m,n=0 and
b − 12 (2b − 1)m+2n (−x)m y n m 4 Ω (m + n) (2b − 1) (b) m! n! m n m,n=0 2 m x y b − 12 ∞ (y − x)n 4 2m+n Ω (3m + n) = (b)m m! n! m,n=0
(2.3)
∞
(b, 2b − 1 ∈ Z− 0 ),
(2.4)
provided that each of the double series involved is absolutely convergent, Z− 0 being the set of nonpositive integers. Proof. If, for convenience, we denote the left-hand side of the first assertion (2.3) by Φ(x, y) and set n −→ n − m (0 m n; m, n ∈ N0 ), we obtain ∞ b − 12 (2b − 1)2m+n x m y n n Φ(x, y) := Ω (m + n) (b) (2b − 1)n m! n! m m,n=0
n ∞ −n, 2b + n − 1, 2 − 2b − n; x y 1 3 − · F = Ω (n) b − , b, − b − n; y 2 n n! 3 2 n=0 2 which, in view of Bailey’s cubic transformation (2.1), yields
∞ (3; −n); 27x y 2 1 (4x + y)n 3 · 3 F2 Ω (n) b − Φ(x, y) = b, − b − n; (4x + y)3 2 n n! n=0 2
K.-Y. Chen et al. / Applied Mathematics Letters 19 (2006) 887–893
n
=
∞ 3
Ω (n)
b−
n=0 m=0
=
∞
Ω (3m + n)
m,n=0
889
1 2 n−m
(b)m b − 12
(x y 2)m (4x + y)n−3m m! (n − 3m)!
n+2m
(b)m
(x y 2 )m (4x + y)n , m! n!
where [κ] denotes the greatest integer in κ ∈ R. We thus complete our proof of the first assertion (2.3) of Theorem 1 under the hypotheses stated already. Our derivation of the second assertion (2.4) of Theorem 1 would similarly make use of Bailey’s cubic transformation (2.2) in place of (2.1). Remark 1. The assertions (2.3) and (2.4) of Theorem 1 are equivalent, since (2.3) (with x −→ 14 y and y −→ −x) is the same as (2.4). The series identities (2.3) (with y = −4x) and (2.4) (with y = x) readily yield the results asserted by Corollary 1 below. be a bounded sequence of complex numbers. Then Corollary 1. Let {Ω (n)}∞ n=0 ∞ b − 12 (2b − 1)2m+n x m (−4x)n n Ω (m + n) (b) (2b − 1)n m! n! m m,n=0 ∞ b − 12 3 n 2n (16x ) (b, 2b − 1 ∈ Z− Ω (3n) = 0) (b) n! n n=0
(2.5)
or, equivalently,
b − 12 (2b − 1)m+2n (−x)m x n m 4 Ω (m + n) (2b − 1) (b) m! n! m n m,n=0 3 m x ∞ b − 12 4 2m (b, 2b − 1 ∈ Z− = Ω (3m) 0 ), (b) m! m m=0 ∞
(2.6)
provided that the series involved converge absolutely. Remark 2. For a fixed k ∈ N0 , we set Ω (n) = δn,3k+1 and Ω (n) = δn,3k+2 in (for example) (2.5), δm,n (m, n ∈ N0 ) being the Kronecker symbol. We thus find from the assertion (2.5) of Corollary 1 that 3k+1 b − 12 (2b − 1)6k−n+2 (−4)n n =0 (2.7) (3k − n + 1)! (b)3k−n+1 (2b − 1)n n! n=0 and 3k+2 n=0
b − 12 (2b − 1)6k−n+4
(−4)n = 0, (3k − n + 2)! (b)3k−n+2 (2b − 1)n n! n
respectively. Furthermore, by setting b = 1 in (2.7) and (2.8), we obtain 3k+1 (−1)n (2n)! (6k − n + 2)! 2 = 0 (n!)3 (3k − n + 1)! n=0
(2.8)
(2.9)
and 3k+2 n=0
(−1)n (2n)! (6k − n + 4)! 2 = 0, (n!)3 (3k − n + 2)!
respectively.
(2.10)
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Remark 3. For a fixed k ∈ N0 , if we put Ω (n) = δn,k , we find from (2.3) that
k 3 b − 12 b − 12 k (2b − 1)k+m x m y k−m 2 m k−3m k−m k−m (x y ) (4x + y) = . (b)m (2b − 1)k−m m! (k − m)! m=0 (b)m m! (k − 3m)! m=0 When y = x and b = 1 in (2.11), we get
k 1 1 k−3m 3 k (k + m)! 2 k−m 2 k−m 5 . 2 = (m!)2 (k − 3m)! m! (k − m)! m=0 m=0
(2.11)
(2.12)
3. Reducible double hypergeometric functions More than three decade ago, Srivastava and Daoust [11] first considered the two-variable case of their multiple hypergeometric function (cf. [12, p. 454]; see also [14, p. 37 et seq.]). For the sake of ready reference, we choose to recall here their definition only in the two-variable case as follows [11, p. 199, Equation (2.1)]:
A:B;B [(a) : ϑ, ϕ] : [(b) : ψ] ; [(b ) : ψ ]; x, y FC:D;D [(c) : ξ, η] : [(d) : ζ ] ; [(d ) : ζ ];
B ∞ A (a ) Bj =1 (b j )nψ j x m y n j =1 j mϑ j +nϕ j j =1 (b j )mψ j := , C D D m! n! m,n=0 j =1 (c j )mξ j +nη j j =1 (d j )mζ j j =1 (d j )nζ
(3.1)
j
where, for convergence of the double hypergeometric series, 1+
C
ξj +
D
j =1
j =1
C
D
ζj −
A
ϑj −
B
j =1
j =1
A
B
ψj ≥ 0
and 1+
j =1
ηj +
j =1
ζ j
−
ϕj −
j =1
j =1
ψ j ≥ 0,
with equality only when |x| and |y| are constrained appropriately (see, for details, [9,13]). Here, for the sake of convenience, (a) abbreviates the array of A parameters a1 , . . . , a A , with similar interpretations for (b), (b ), (c), (d), and (d ). Many useful cases of reducibility of double hypergeometric functions are known to exist when the parameters ϑ, ϕ, ψ, ψ , ξ, η, ζ , and ζ in (3.1) are chosen to be 1 (see, for example, [5, pp. 439–440]; see also [14, pp. 28–32]). By setting b = λ and Ω (n) =
Aj=1 (a j )n Bj=1 (b j )n
(n ∈ N0 )
(3.2)
in Theorem 1 and Corollary 1, we obtain several presumably new transformation and reduction formulas for the Srivastava–Daoust hypergeometric function in two variables. The resulting hypergeometric transformation and reduction formulas are being recorded here. Theorem 2. The following transformation formula holds true for the Srivastava–Daoust double hypergeometric function defined by (3.1): 1 ; λ− :1 ; A+1:0;1 [2λ − 1 : 2, 1], [(a) : 1, 1]: x, y FB:1;1 2 [(b) : 1, 1]: [λ : 1] ; [2λ − 1 : 1]; 1 : 2, 1 , [(a) : 3, 1] : ; ; 2 A+1:0;0 λ − = FB:1;0 x y , 4x + y (3.3) 2 [(b) : 3, 1] : [λ : 1] ; ;
K.-Y. Chen et al. / Applied Mathematics Letters 19 (2006) 887–893
891
or, equivalently, 1 y ; :1 ; A+1:1;0 [2λ − 1 : 1, 2], [(a) : 1, 1]: λ − − x, FB:1;1 2 4 [(b) : 1, 1]: [2λ − 1 : 1] ; [λ : 1] ; 1 2y x : 2, 1 , [(a) : 3, 1] : ; ; A+1:0;0 λ − , y − x . = FB:1;0 2 ; 4 [(b) : 3, 1] : [λ : 1] ;
(3.4)
Corollary 2. The following hypergeometric reduction formula holds true: 1 : 1 ; [2λ − 1 : 2, 1], [(a) : 1, 1]: ; λ − A+1:0;1 FB:1;1 x, −4x 2 [(b) : 1, 1]: [λ : 1] ; [2λ − 1 : 1];
1 , [3; (a)]; 2; λ − = 3 A+2 F 3B+1 (3 A−B · 4x)3 2 λ, [3; (b)]; or, equivalently, 1 λ− :1 ; A+1:1;0 [2λ − 1 : 1, 2], [(a) : 1, 1]: FB:1;1 2 [(b) : 1, 1]:[2λ − 1 : 1]; [λ : 1] ;
1 , [3; (a)]; 2; λ − (3 A−B · x)3 . = 3 A+2 F 3B+1 2 λ, [3; (b)];
;
(3.5)
x − x, 4 (3.6)
4. Associated generating-function relationships In this section, we show how one of the double-series identities (asserted by Theorem 1) can be applied to derive the hypergeometric generating-function relationships (4.1) and (4.2) below (see also [15] for various other families of generating-function relationships for hypergeometric polynomials). Theorem 3. The following hypergeometric generating-function relationships hold true: 1 ∞ λ − Aj=1 (a j )n {(1 − x)t}n 2 n
n=0
Bj=1 (b j )n
n!
(N; −n), [N − 1; 1 − (b) − n], (N; 2 − 2λ − n); 3 z · 2N+(N−1)B F 2N+(N−1) A−2 [N − 1; 1 − (a) − n], (N; − λ − n), (N − 2; 2 − 2λ − n), λ; 2 1 ; λ− :1 ; N A+1:0;1 [2λ − 1 : 2, 1], [(a) : 1, 1] : xt , (1 − x)t = FB:1;1 2 [(b) : 1, 1] : [λ : 1] ; [2λ − 1 : 1]; 1 : 2, 1 , [(a) : 3, 1] : ; ; A+1:0;0 λ − = FB:1;0 x(1 − x)2 t N+2 , (1 − x)t + 4xt N 2 [(b) : 3, 1] : [λ : 1] ; ; (N ∈ N \ {1, 2}), where, for convenience, z := (−1)(N−1)( A+B)
x N N (N − 1)(N−1)(B−A) . (N − 2) N−2 (1 − x) N
(4.1)
(4.2)
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K.-Y. Chen et al. / Applied Mathematics Letters 19 (2006) 887–893
Proof. Let Ψ (x, t) denote the first member of the hypergeometric generating-function relationship (4.1). Then, upon replacing the hypergeometric polynomial in Ψ (x, t) by its series expression and simplifying the various Pochhammer symbols involved, we obtain n ∞ λ − 12 Aj=1 (a j )n {(1 − x)t}n N (−n) Nm (2 − 2λ − n) Nm n Ψ (x, t) = B 3 n! j =1 (b j )n (λ)m n=0 m=0 m!(2 − 2λ − n)(N−2)m 2 − λ − n Nm
m Bj=1 (1 − b j − n)(N−1)m x · A (−1)(N−1)( A+B) (1 − x) N j =1 (1 − a j − n)(N−1)m n 1 λ − ∞ (2λ − 1)n−(N−2)m A (a j )n−(N−1)m N n −N m 2 {(1 − x)t} {x(1 − x) } j =1 n−Nm = B (b ) (n − Nm)! m! (λ) (2λ − 1) m n−Nm j =1 j n−(N−1)m n=0 m=0 =
∞ (λ − 12 )n (2λ − 1)2m+n Aj=1 (a j )m+n (xt N )m {(1 − x)t}n . (λ)m (2λ − 1)n n! Bj=1 (b j )m+n m! m,n=0
(4.3)
The second member of the hypergeometric generating-function relationship (4.1) would result on appealing to the definition (3.1). Alternatively, the hypergeometric generating-function relationship (4.1) can be deduced by suitably specializing Srivastava’s result which was subsequently reproduced in the monograph on the subject by Srivastava and Manocha [15, p. 144, Equation 2.6 (28)]. Next, by specializing the sequence {Ω (n)}∞ n=0 just as in (3.2), and by setting b = λ, x −→ xt N ,
and
y −→ (1 − x)t,
in the assertion (2.3) of Theorem 1, we find from (4.3) that 1 A λ − ∞ 2 2m+n j =1 (a j )3m+n {x(1 − x)2 t N+2 }m {(1 − x)t + 4xt N }n Ψ (x, t) = (λ)m m! n! Bj=1 (b j )3m+n m,n=0 1 : 2, 1 , [(a) : 3, 1] : ; ; A+1:0;0 λ − x(1 − x)2 t N+2 , (1 − x)t + 4xt N , = FB:1;0 2 [(b) : 3, 1] : [λ : 1]; ; which evidently proves the assertion (4.2) of Theorem 3.
Acknowledgements The present investigation was supported, in part, by the National Science Council of the Republic of China under Grant NSC 93-2115-M-032-011 and the Natural Sciences and Engineering Research Council of Canada under Grant OGP0007353. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
G.E. Andrews, R. Askey, R. Roy, Special Functions, Cambridge University Press, Cambridge, London, New York, 1999. R. Askey, A look at the Bateman project, Contemp. Math. 169 (1994) 29–43. W.N. Bailey, Products of generalized hypergeometric series, Proc. London Math. Soc. (2) 28 (1928) 242–254. W.N. Bailey, Transformations of generalized hypergeometric series, Proc. London Math. Soc. (2) 29 (1929) 495–502. R.G. Buschman, H.M. Srivastava, Series identities and reducibility of Kamp´e de F´eriet functions, Math. Proc. Cambridge Philos. Soc. 91 (1982) 435–440. K.-Y. Chen, H.M. Srivastava, Series identities and associated families of generating functions, J. Math. Anal. Appl. 311 (2005) 582–599. A. Erd´elyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions, vol. I, McGraw-Hill Book Company, New York, Toronto, London, 1953. B. Gonz´alez, J. Matera, H.M. Srivastava, Some q-generating functions and associated families of generalized hypergeometric polynomials, Math. Comput. Modelling 34 (1–2) (2001) 133–175. N.T. H`ai, O.I. Marichev, H.M. Srivastava, A note on the convergence of certain families of multiple hypergeometric series, J. Math. Anal. Appl. 164 (1992) 104–115.
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[10] E.D. Rainville, Special Functions, Macmillan Company, New York, 1960. Reprinted by Chelsea Publishing Company, Bronx, New York, 1971. [11] H.M. Srivastava, M.C. Daoust, On Eulerian integrals associated with Kamp´e de F´eriet’s function, Publ. Inst. Math. (Beograd) (N.S.) 9 (23) (1969) 199–202. [12] H.M. Srivastava, M.C. Daoust, Certain generalized Neumann expansions associated with the Kamp´e de F´eriet function, Nederl. Akad. Wetensch. Indag. Math. 31 (1969) 449–457. [13] H.M. Srivastava, M.C. Daoust, A note on the convergence of Kamp´e de F´eriet’s double hypergeometric series, Math. Nachr. 53 (1972) 151–159. [14] H.M. Srivastava, P.W. Karlsson, Multiple Gaussian Hypergeometric Series, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane, Toronto, 1985. [15] H.M. Srivastava, H.L. Manocha, A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane, Toronto, 1984.
Some double-series identities and associated generating-function relationships Kung-Yu Chen a , Shuoh-Jung Liu a , H.M. Srivastava b,∗ a Department of Mathematics, Tamkang University, Tamsui 25137, Taiwan, ROC b Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3P4, Canada
Received 8 July 2005; accepted 22 July 2005
Abstract The main object of the present work is to investigate several families of double-series identities and their applications to the Srivastava–Daoust hypergeometric function in two variables. A number of associated generating-function relationships, involving certain classes of hypergeometric polynomials, are also considered. c 2005 Elsevier Ltd. All rights reserved. Keywords: Series identities; Srivastava–Daoust multivariable hypergeometric function; Generating-function relationships; Hypergeometric functions and polynomials; Cubic transformations; Generating functions
1. Introduction and definitions As usual, we denote by p F q a generalized hypergeometric function with p numerator and q denominator parameters (see, for details, [7,10]). Also, throughout this work, (λ)ν is the Pochhammer symbol (or the shifted factorial, since (1)n = n! for n ∈ N0 ) defined (for λ, ν ∈ C and in terms of the familiar Gamma function) by Γ (λ + ν) 1 (ν = 0; λ ∈ C \ {0}) = (λ)ν := λ(λ + 1) · · · (λ + n − 1) (ν = n ∈ N; λ ∈ C), Γ (λ) where N0 := N ∪ {0} (N := {1, 2, 3, . . .}). For m ∈ N, we find it to be convenient to abbreviate the array of m parameters λ+m−1 λ λ+1 , ,..., m m m simply by (m; λ), the array being understood to be empty when m = 0. ∗ Corresponding author. Tel.: +1 250 721 7455; fax: +1 250 721 8962.
E-mail addresses: [email protected] (K.-Y. Chen), [email protected] (S.-J. Liu), [email protected] (H.M. Srivastava). c 2005 Elsevier Ltd. All rights reserved. 0893-9659/$ - see front matter doi:10.1016/j.aml.2005.07.013
888
K.-Y. Chen et al. / Applied Mathematics Letters 19 (2006) 887–893
Motivated essentially by several recent investigations dealing with (for example) series identities and their various applications (cf. [6,8]; see also [5] and the other references cited in each of these earlier works), we prove two doubleseries identities and apply these identities in order to deduce hypergeometric reduction formulas and generatingfunction relationships associated with the Srivastava–Daoust hypergeometric function in two variables, which is defined in Section 3 below. 2. Series identities based upon Bailey’s cubic transformations Over seven decades ago, the following two cubic transformations were proven by Wilfrid Norman Bailey (1893–1961) (cf. [3,4]; see also [2]): a, 2b − a − 1, a − 2b + 2; x (3; a); 27x 3 3 − (2.1) = (1 − x)−a 3 F 2 3F2 b, a − b + ; 4 b, a − b + ; 4(1 − x)3 2 2 and 1 (3; a); 27x 2 x −a a, b − , a − b + 1; 3 . (2.2) x = 1− 2 3F2 3 F2 b, a − b + ; (4 − x)3 4 2b − 1, 2a − 2b + 2; 2 Bailey’s cubic transformation (2.2) is recorded erroneously by Andrews et al. [1, p. 185, Exercise 38]. Making use of (2.1) and (2.2), we first prove Theorem 1 below. Theorem 1. Let {Ω (n)}∞ n=0 be a bounded sequence of complex numbers. Then 1 ∞ b − 2 n (2b − 1)2m+n x m y n Ω (m + n) (b)m (2b − 1)n m! n! m,n=0 b − 12 ∞ 2 m n 2m+n (x y ) (4x + y) (b, 2b − 1 ∈ Z− Ω (3m + n) = 0) (b) m! n! m m,n=0 and
b − 12 (2b − 1)m+2n (−x)m y n m 4 Ω (m + n) (2b − 1) (b) m! n! m n m,n=0 2 m x y b − 12 ∞ (y − x)n 4 2m+n Ω (3m + n) = (b)m m! n! m,n=0
(2.3)
∞
(b, 2b − 1 ∈ Z− 0 ),
(2.4)
provided that each of the double series involved is absolutely convergent, Z− 0 being the set of nonpositive integers. Proof. If, for convenience, we denote the left-hand side of the first assertion (2.3) by Φ(x, y) and set n −→ n − m (0 m n; m, n ∈ N0 ), we obtain ∞ b − 12 (2b − 1)2m+n x m y n n Φ(x, y) := Ω (m + n) (b) (2b − 1)n m! n! m m,n=0
n ∞ −n, 2b + n − 1, 2 − 2b − n; x y 1 3 − · F = Ω (n) b − , b, − b − n; y 2 n n! 3 2 n=0 2 which, in view of Bailey’s cubic transformation (2.1), yields
∞ (3; −n); 27x y 2 1 (4x + y)n 3 · 3 F2 Ω (n) b − Φ(x, y) = b, − b − n; (4x + y)3 2 n n! n=0 2
K.-Y. Chen et al. / Applied Mathematics Letters 19 (2006) 887–893
n
=
∞ 3
Ω (n)
b−
n=0 m=0
=
∞
Ω (3m + n)
m,n=0
889
1 2 n−m
(b)m b − 12
(x y 2)m (4x + y)n−3m m! (n − 3m)!
n+2m
(b)m
(x y 2 )m (4x + y)n , m! n!
where [κ] denotes the greatest integer in κ ∈ R. We thus complete our proof of the first assertion (2.3) of Theorem 1 under the hypotheses stated already. Our derivation of the second assertion (2.4) of Theorem 1 would similarly make use of Bailey’s cubic transformation (2.2) in place of (2.1). Remark 1. The assertions (2.3) and (2.4) of Theorem 1 are equivalent, since (2.3) (with x −→ 14 y and y −→ −x) is the same as (2.4). The series identities (2.3) (with y = −4x) and (2.4) (with y = x) readily yield the results asserted by Corollary 1 below. be a bounded sequence of complex numbers. Then Corollary 1. Let {Ω (n)}∞ n=0 ∞ b − 12 (2b − 1)2m+n x m (−4x)n n Ω (m + n) (b) (2b − 1)n m! n! m m,n=0 ∞ b − 12 3 n 2n (16x ) (b, 2b − 1 ∈ Z− Ω (3n) = 0) (b) n! n n=0
(2.5)
or, equivalently,
b − 12 (2b − 1)m+2n (−x)m x n m 4 Ω (m + n) (2b − 1) (b) m! n! m n m,n=0 3 m x ∞ b − 12 4 2m (b, 2b − 1 ∈ Z− = Ω (3m) 0 ), (b) m! m m=0 ∞
(2.6)
provided that the series involved converge absolutely. Remark 2. For a fixed k ∈ N0 , we set Ω (n) = δn,3k+1 and Ω (n) = δn,3k+2 in (for example) (2.5), δm,n (m, n ∈ N0 ) being the Kronecker symbol. We thus find from the assertion (2.5) of Corollary 1 that 3k+1 b − 12 (2b − 1)6k−n+2 (−4)n n =0 (2.7) (3k − n + 1)! (b)3k−n+1 (2b − 1)n n! n=0 and 3k+2 n=0
b − 12 (2b − 1)6k−n+4
(−4)n = 0, (3k − n + 2)! (b)3k−n+2 (2b − 1)n n! n
respectively. Furthermore, by setting b = 1 in (2.7) and (2.8), we obtain 3k+1 (−1)n (2n)! (6k − n + 2)! 2 = 0 (n!)3 (3k − n + 1)! n=0
(2.8)
(2.9)
and 3k+2 n=0
(−1)n (2n)! (6k − n + 4)! 2 = 0, (n!)3 (3k − n + 2)!
respectively.
(2.10)
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Remark 3. For a fixed k ∈ N0 , if we put Ω (n) = δn,k , we find from (2.3) that
k 3 b − 12 b − 12 k (2b − 1)k+m x m y k−m 2 m k−3m k−m k−m (x y ) (4x + y) = . (b)m (2b − 1)k−m m! (k − m)! m=0 (b)m m! (k − 3m)! m=0 When y = x and b = 1 in (2.11), we get
k 1 1 k−3m 3 k (k + m)! 2 k−m 2 k−m 5 . 2 = (m!)2 (k − 3m)! m! (k − m)! m=0 m=0
(2.11)
(2.12)
3. Reducible double hypergeometric functions More than three decade ago, Srivastava and Daoust [11] first considered the two-variable case of their multiple hypergeometric function (cf. [12, p. 454]; see also [14, p. 37 et seq.]). For the sake of ready reference, we choose to recall here their definition only in the two-variable case as follows [11, p. 199, Equation (2.1)]:
A:B;B [(a) : ϑ, ϕ] : [(b) : ψ] ; [(b ) : ψ ]; x, y FC:D;D [(c) : ξ, η] : [(d) : ζ ] ; [(d ) : ζ ];
B ∞ A (a ) Bj =1 (b j )nψ j x m y n j =1 j mϑ j +nϕ j j =1 (b j )mψ j := , C D D m! n! m,n=0 j =1 (c j )mξ j +nη j j =1 (d j )mζ j j =1 (d j )nζ
(3.1)
j
where, for convergence of the double hypergeometric series, 1+
C
ξj +
D
j =1
j =1
C
D
ζj −
A
ϑj −
B
j =1
j =1
A
B
ψj ≥ 0
and 1+
j =1
ηj +
j =1
ζ j
−
ϕj −
j =1
j =1
ψ j ≥ 0,
with equality only when |x| and |y| are constrained appropriately (see, for details, [9,13]). Here, for the sake of convenience, (a) abbreviates the array of A parameters a1 , . . . , a A , with similar interpretations for (b), (b ), (c), (d), and (d ). Many useful cases of reducibility of double hypergeometric functions are known to exist when the parameters ϑ, ϕ, ψ, ψ , ξ, η, ζ , and ζ in (3.1) are chosen to be 1 (see, for example, [5, pp. 439–440]; see also [14, pp. 28–32]). By setting b = λ and Ω (n) =
Aj=1 (a j )n Bj=1 (b j )n
(n ∈ N0 )
(3.2)
in Theorem 1 and Corollary 1, we obtain several presumably new transformation and reduction formulas for the Srivastava–Daoust hypergeometric function in two variables. The resulting hypergeometric transformation and reduction formulas are being recorded here. Theorem 2. The following transformation formula holds true for the Srivastava–Daoust double hypergeometric function defined by (3.1): 1 ; λ− :1 ; A+1:0;1 [2λ − 1 : 2, 1], [(a) : 1, 1]: x, y FB:1;1 2 [(b) : 1, 1]: [λ : 1] ; [2λ − 1 : 1]; 1 : 2, 1 , [(a) : 3, 1] : ; ; 2 A+1:0;0 λ − = FB:1;0 x y , 4x + y (3.3) 2 [(b) : 3, 1] : [λ : 1] ; ;
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or, equivalently, 1 y ; :1 ; A+1:1;0 [2λ − 1 : 1, 2], [(a) : 1, 1]: λ − − x, FB:1;1 2 4 [(b) : 1, 1]: [2λ − 1 : 1] ; [λ : 1] ; 1 2y x : 2, 1 , [(a) : 3, 1] : ; ; A+1:0;0 λ − , y − x . = FB:1;0 2 ; 4 [(b) : 3, 1] : [λ : 1] ;
(3.4)
Corollary 2. The following hypergeometric reduction formula holds true: 1 : 1 ; [2λ − 1 : 2, 1], [(a) : 1, 1]: ; λ − A+1:0;1 FB:1;1 x, −4x 2 [(b) : 1, 1]: [λ : 1] ; [2λ − 1 : 1];
1 , [3; (a)]; 2; λ − = 3 A+2 F 3B+1 (3 A−B · 4x)3 2 λ, [3; (b)]; or, equivalently, 1 λ− :1 ; A+1:1;0 [2λ − 1 : 1, 2], [(a) : 1, 1]: FB:1;1 2 [(b) : 1, 1]:[2λ − 1 : 1]; [λ : 1] ;
1 , [3; (a)]; 2; λ − (3 A−B · x)3 . = 3 A+2 F 3B+1 2 λ, [3; (b)];
;
(3.5)
x − x, 4 (3.6)
4. Associated generating-function relationships In this section, we show how one of the double-series identities (asserted by Theorem 1) can be applied to derive the hypergeometric generating-function relationships (4.1) and (4.2) below (see also [15] for various other families of generating-function relationships for hypergeometric polynomials). Theorem 3. The following hypergeometric generating-function relationships hold true: 1 ∞ λ − Aj=1 (a j )n {(1 − x)t}n 2 n
n=0
Bj=1 (b j )n
n!
(N; −n), [N − 1; 1 − (b) − n], (N; 2 − 2λ − n); 3 z · 2N+(N−1)B F 2N+(N−1) A−2 [N − 1; 1 − (a) − n], (N; − λ − n), (N − 2; 2 − 2λ − n), λ; 2 1 ; λ− :1 ; N A+1:0;1 [2λ − 1 : 2, 1], [(a) : 1, 1] : xt , (1 − x)t = FB:1;1 2 [(b) : 1, 1] : [λ : 1] ; [2λ − 1 : 1]; 1 : 2, 1 , [(a) : 3, 1] : ; ; A+1:0;0 λ − = FB:1;0 x(1 − x)2 t N+2 , (1 − x)t + 4xt N 2 [(b) : 3, 1] : [λ : 1] ; ; (N ∈ N \ {1, 2}), where, for convenience, z := (−1)(N−1)( A+B)
x N N (N − 1)(N−1)(B−A) . (N − 2) N−2 (1 − x) N
(4.1)
(4.2)
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Proof. Let Ψ (x, t) denote the first member of the hypergeometric generating-function relationship (4.1). Then, upon replacing the hypergeometric polynomial in Ψ (x, t) by its series expression and simplifying the various Pochhammer symbols involved, we obtain n ∞ λ − 12 Aj=1 (a j )n {(1 − x)t}n N (−n) Nm (2 − 2λ − n) Nm n Ψ (x, t) = B 3 n! j =1 (b j )n (λ)m n=0 m=0 m!(2 − 2λ − n)(N−2)m 2 − λ − n Nm
m Bj=1 (1 − b j − n)(N−1)m x · A (−1)(N−1)( A+B) (1 − x) N j =1 (1 − a j − n)(N−1)m n 1 λ − ∞ (2λ − 1)n−(N−2)m A (a j )n−(N−1)m N n −N m 2 {(1 − x)t} {x(1 − x) } j =1 n−Nm = B (b ) (n − Nm)! m! (λ) (2λ − 1) m n−Nm j =1 j n−(N−1)m n=0 m=0 =
∞ (λ − 12 )n (2λ − 1)2m+n Aj=1 (a j )m+n (xt N )m {(1 − x)t}n . (λ)m (2λ − 1)n n! Bj=1 (b j )m+n m! m,n=0
(4.3)
The second member of the hypergeometric generating-function relationship (4.1) would result on appealing to the definition (3.1). Alternatively, the hypergeometric generating-function relationship (4.1) can be deduced by suitably specializing Srivastava’s result which was subsequently reproduced in the monograph on the subject by Srivastava and Manocha [15, p. 144, Equation 2.6 (28)]. Next, by specializing the sequence {Ω (n)}∞ n=0 just as in (3.2), and by setting b = λ, x −→ xt N ,
and
y −→ (1 − x)t,
in the assertion (2.3) of Theorem 1, we find from (4.3) that 1 A λ − ∞ 2 2m+n j =1 (a j )3m+n {x(1 − x)2 t N+2 }m {(1 − x)t + 4xt N }n Ψ (x, t) = (λ)m m! n! Bj=1 (b j )3m+n m,n=0 1 : 2, 1 , [(a) : 3, 1] : ; ; A+1:0;0 λ − x(1 − x)2 t N+2 , (1 − x)t + 4xt N , = FB:1;0 2 [(b) : 3, 1] : [λ : 1]; ; which evidently proves the assertion (4.2) of Theorem 3.
Acknowledgements The present investigation was supported, in part, by the National Science Council of the Republic of China under Grant NSC 93-2115-M-032-011 and the Natural Sciences and Engineering Research Council of Canada under Grant OGP0007353. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
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[10] E.D. Rainville, Special Functions, Macmillan Company, New York, 1960. Reprinted by Chelsea Publishing Company, Bronx, New York, 1971. [11] H.M. Srivastava, M.C. Daoust, On Eulerian integrals associated with Kamp´e de F´eriet’s function, Publ. Inst. Math. (Beograd) (N.S.) 9 (23) (1969) 199–202. [12] H.M. Srivastava, M.C. Daoust, Certain generalized Neumann expansions associated with the Kamp´e de F´eriet function, Nederl. Akad. Wetensch. Indag. Math. 31 (1969) 449–457. [13] H.M. Srivastava, M.C. Daoust, A note on the convergence of Kamp´e de F´eriet’s double hypergeometric series, Math. Nachr. 53 (1972) 151–159. [14] H.M. Srivastava, P.W. Karlsson, Multiple Gaussian Hypergeometric Series, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane, Toronto, 1985. [15] H.M. Srivastava, H.L. Manocha, A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane, Toronto, 1984.