R E S E A R C H
Open Access
Certain transformations for multiple
hypergeometric functions
Chuanan Wei
1, Xiaoxia Wang
2*and Yaqiong Li
2*Correspondence:
[email protected] 2Department of Mathematics,
Shanghai University, Shanghai, 200444, P.R. China
Full list of author information is available at the end of the article
Abstract
The aim of this research is to provide many new hypergeometric transformations of Kampé de Fériet’s double hypergeometric functions and Srivastava’s triple
hypergeometric functions, which are obtained with the help of the relations of Appell, Humbert, Horn and Srivastava’s multiple hypergeometric functions. The results are derived by the beta integral method which has been recently introduced and studied systematically by Krattenthaler and Rao.
MSC: 33C20; 33C05; 33B20
Keywords: Appell functions; Srivastava functions; multiple hypergeometric series; beta integral
1 Introduction
The generalized hypergeometric functionpFq withpnumerator andqdenominator
pa-rameters may be defined by []
pFq
α, . . . ,αp;
β, . . . ,βq;
z
=pFq[α, . . . ,αp;β, . . . ,βq;z]
=
n≥
(α)n· · ·(αp)n
(β)n· · ·(βq)n
·zn
n!, (.)
where (α)ndenotes the Pochhammer symbol (or the shifted factorial, since ()n=n!)
de-fined, for any complex numberα, by
(α)n=
(α+n)
(α) =
⎧ ⎨ ⎩
α(α+ )· · ·(α+n– ), n∈N= , , , . . . ,
, n= , (.)
whereis the well-known gamma function. For convergence condition and properties of
pFq, we refer to [, ]. ThepFqfunction extends the Gauss hypergeometric function
F
A,B;
C; z
=
n≥
(A)n(B)n
(C)n
zn
n!
by increasing the number of parameters in the numerator as well as in the denominator. On the other hand, the notation defined and introduced by Kampé de Fériet for his dou-ble hypergeometric function of superior order was subsequently abbreviated by Burchnall
and Chaundy []. We, however, recall here the definition of a more general (than the one defined by Kampé de Fériet) double hypergeometric function in a slightly modified no-tation given by Srivastava and Panda [, p., Eq. ()]. For this, let (Hh) denote the
se-quence of parameters (H,H, . . . ,Hh), and let nonnegative integers define the
Pochham-mer symbols ((Hh))n= (H)n(H)n· · ·(Hh)n, where, whenn= , the product is understood
to reduce to unity. Therefore, the convenient generalization of the Kampé de Fériet func-tion is defined as follows:
Fgh::ca;;db
(Hh) : (Aa); (Bb);
(Gg) : (Cc); (Dd);
x,y
=
m,n≥
((Hh))m+n((Aa))m((Bb))n
((Gg))m+n((Cc))m((Dd))n
xm
m! yn
n!.
For more details about the convergence of this function, we refer to []. Various authors (see,e.g., [–]) have discussed the reducibility of the Kampé de Fériet function.
Finally, we introduce Srivastava’s [, p.] triple hypergeometric seriesF()[x,y,z] de-fined by
F()[x,y,z]≡F()
(a) :: (b); (b); (b) : (c); (c); (c); (e) :: (g); (g); (g) : (h); (h); (h); x,y,z
=
∞
m,n,p=
(m,n,p)x
m
m! yn
n! zp
p!,
where, for convenience,
(m,n,p) =
A
j=(aj)m+n+p
B j=(bj)m+n
B j=(bj)n+p
B j=(bj)m+p
E
j=(ej)m+n+p
G j=(gj)m+n
G j=(gj)n+p
G j=(gj)m+p
·
C j=(cj)m
C j=(cj)n
C j=(cj)p
H j=(hj)m
H j=(hj)n
H j=(hj)p
,
and (a) abbreviates the array ofAparametersa, . . . ,aA, with similar interpretations for
(b), (b), (b), and so on.
The reduction formulas of the general hypergeometric functions in one, two and more variables have been investigated, and some of them can be seen in [, ].
In addition, the beta function B(α,β) is defined by the first integral and known to be evaluated as the second one as follows:
B(α,β) =
⎧ ⎨ ⎩
t
α–( –t)β–dt (Re(α) > ;Re(β) > );
(α)(β)
(α+β) (α,β∈C\Z – ).
(.)
Recently, Krattenthaler and Rao [] (see also []) made a systematic use of the so-called beta integral method, a method of derivingnewhypergeometric identities fromoldones by mainly using the beta integral in (.) based on the Mathematica Package HYP, to illus-trate several interesting identities for the hypergeometric functions and Kampé de Fériet functions.
Kampé de Fériet’s double hypergeometric function and Srivastava’s triple hypergeometric functions by employing the so-called beta integral method. Moreover, in our work, we use the beta integral to establish transformations two and three times, and this is a general-ization of Krattenthaler and Rao’s method.
2 The new transformations from Appell functions
The following four functions are Appell functions [, ] which are famous in the field of the hypergeometric functions [, , ]:
F[a;b,b;c;x,y] :=
m,n≥
(a)m+n(b)m(b)n
(c)m+n
xm
m! yn
n!;
F[a;b,b;c,c;x,y] :=
m,n≥
(a)m+n(b)m(b)n
(c)m(c)n
xm
m! yn
n!;
F[a,a;b,b;c;x,y] :=
m,n≥
(a)m(a)n(b)m(b)n
(c)m+n
xm
m! yn
n!;
F[a;b;c,c;x,y] :=
m,n≥
(a)m+n(b)m+n
(c)m(c)n
xm
m! yn
n!.
In this part, we present the transformations which are obtained from Appell’s four hy-pergeometric functions and we just take two of them as examples and the others can be derived in the same manner.
We give two theorems as examples in this part.
Theorem Assuming b or b is a nonpositive integer,and d and e are not nonpositive
integers,the following result holds true:
F:;:;
a:b,e+b;b,d+b–e; c,d+b+b: –; –; ,
=
e,d–e,d+b+b d,e+b,d+b–e
m,n≥
(c–a)m+n(e)m–n(b)m(b)n
(c)m+n( +e–d)m–nm!n!
.
It is interesting that the series on the right-hand side of the assertion of Theorem is similar to Horn’sGandH[, pp.-].
Proof We start with the following well-known transformation of Appell functionF[, p., Eq. (..)]:
F[a;b,b;c;x,y] = ( –x)–b( –y)–bF
c–a;b,b;c; x x– ,
y y–
. (.)
Settingy= –xin (.), we get the following identity ofx:
F[a;b,b;c;x, –x] = ( –x)–bx–bF
c–a;b,b;c; x x– ,
–x –x
. (.)
in-terchanging the order of integration and summation, which is easily seen to be justified due to the uniform convergence of the series in the interval (, ) involved in the pro-cess, evaluating the integral on both sides using the beta integral (.), then after some simplification, summing up the series on both sides, we easily arrive at the assertion of
Theorem .
Theorem Assuming bor bis a nonpositive integer,and d,d,eand eare not
non-positive integers,the following result holds true:
F:;:;
a:b,e;b,e; c:b+d;b+d; ,
=
d–e,d–e,b+d,b+d d,d,b+d–e,b+d–e
·F:;:;
c–a:b,e;b,e;
c: +e–d; +e–d; ,
. (.)
Proof We prove Theorem by applying the beta integral method twice. Now, multiply-ing both sides of equation (.) byxe–( –x)b+d–e–ye–( –x)b+d–e–, integrating the resulting equation with respect toxandyboth from to , expressing the involvedFas series, interchanging the order of integration and summation, which is easily seen to be justified due to the uniform convergence of the series in the interval (, ) involved in the process, evaluating the integral on both sides using the beta integral (.) twice, then after some simplification, summing up the series on both sides, we easily arrive at the assertion
of Theorem .
In the following part, all the transformations are obtained by the beta integral method and the derivations are the same as in the proof of Theorems and . We will just present the results without their proofs.
If we start with the formula [, Eq. ()] withy=x,
Fa;b;c, +a+b–c;x( –x),x( –x)=F
a,b;
c; x
F
a,b; +a+b–c; x
,
we have the following transformation:
F:;:;
a,b,e,d–e: –; –;
d
,
d+
:c; +a+b–c; ,
=F:;:;
e:a,b;a,b;
d:c; +a+b–c; ,
(.)
withebeing a nonpositive integer.
We start with the formula [, Eq. ()] withy= –x,
Fa;b;c, +a+b–c;x, ( –x)=F
a,b;
c; x
F
a,b;
+a+b–c; –x
we have
Especially, the right-hand side of (.) is equal to the left-hand side of the result [, Eq. (.)] by Krattenthaler and Rao:
F:;:;
Comparing the identities (.) with (.), we have a new relation between the double hy-pergeometric function and the generalized hyhy-pergeometric functionF:
F:;:;
If we start with the transformation [, Eq. ()],
F
applying the beta integral method twice, after some simplification, we can get the following transformation:
whereais a nonpositive integer.
If we start with the transformation [, Eq. ()] withx→ux,
In fact, this result can also be derived by the following Gauss summation theorem [, §.]
F
If we start with the equation [, Eq. ()] withy= –x,
we get the following transformation:
F:;:;
and the double series on the left-hand side of the above identity can be considered as a generalization of Horn’sH[].
If we start with the transformation [, Eq. ()] withy→–y,
we have, after applying the beta integral twice,
F:;:;
The double hypergeometric series on the right-hand side of the above identity can be con-sidered as a generalization of Horn’sH[] and is similar to Exton’sX[].
Substitutingyby –xin the result [, Eq. ()], we have
We then obtain the following transformation by the beta integral method:
F:;:;
whereais a nonpositive integer.
and assume thatais a nonpositive integer, we have
and providedais a nonpositive integer, we have
F:;:;
and assume thatais a nonpositive integer, we obtain the following identity:
F:;:;
we obtain the following transformation by using the beta integral method onx:
F:;:;
whereais a nonpositive integer.
Remark In fact, there are many other identities of Appell functions which can be used to
establish the new transformations. Here we just illustrate three results.
3 The transformations from Horn’s functionsG1andG2
If we start with the transformation betweenFandGin [, Eq. (.)] (see also []) withy→–xandx→–x,
G(α,α;β,β; –x, –x)
= ( –x)–α–αF
–β–β;α,α; –β, –β; –x –x,
–x –x
,
we obtain
F:;:;
e, –β–β:α;α; +e–d: –β; –β;
,
=
d,α+α+d–e d–e,α+α+d
m,n≥
(e)m+n(β)n–m(β)m–n(α)m(α)n
(–)m+n(α
+α+d)m+nm!n!
,
whereeis a nonpositive integer. In fact, the double hypergeometric function on the right-hand side of the above identity can be considered as a generalization ofG.
If we start with the transformation betweenFandGin [, Eq. (.)] (see also []) withy= –x,
G(α;β,β;x, –x) = ( –x)–αF
α, –β–β; –β, –β; x –x,
–x –x
,
and provided thataoreis a nonpositive integer, we have
F:;:;
e,α, –β–β: –; –; +e–d: –β; –β;
–,
=
d,α+d–e d–e,α+d
m,n≥
(e)m+n(α)m+n(β)n–m(β)m–n
(α+d)m+nm!n!
(–)n.
The double hypergeometric function on the right-hand side of the above identity can be considered as a generalization ofG.
4 The transformations from Humbert’s functions
2,
2and
3
First, we present Humbert’s three functions defined by (see [, ])
(b;c;x,y) =
m,n≥ (b)m
(c)m+n
xmyn
m!n!,
(a;b,c;x,y) =
m,n≥ (a)m+n
(b)m(c)n
xmyn
m!n!,
(a,b;c;x,y) =
m,n≥
(a)m(b)n
(c)m+n
xmyn
m!n!.
The above series converge absolutely for anyx,y∈C.
If we start with the transformation [, Eq. ()] withx→uxandy→–ux,
(b;b,c;ux, –ux) =
we have
Also, the right-hand side of the above identity can be considered as a generalization of Horn’sH [] and is similar to Exton’s X []. The special case ofb=cin the above relation can be written as
F:;:;
This identity is a relation between general and double hypergeometric functions. If we start with the transformation [, Eq. ()] withx→uxandy→u–ux,
The special case of the above identity withb=cis a relation between a double hypergeo-metric function and a general hypergeohypergeo-metric function
F:;:;
This relation converges for everyu∈C.
If we start with the equation [, Eq. ()] withx→ux,
Comparing the identities (.) with (.), we can obtain a new transformation
F
5 The relations from triple hypergeometric functionsHAandHB The following functions are given by Srivastava [–]:
Multiplying both sides of the above transformation by
xe–( –x)d–e–ye–( –y)d+β–e–ze–( –z)d+α–e–
and applying the beta integral three times with respect tox, y, z, we get the following transformation using the same technique in Theorems and .
F()
– ::β; –;α:e;e;e; – :: –; –; – :γ,d;d+β;d+α;
, ,
=
d–e,d–e,d+β,d+α d,d,d+β–e,d+α–e
F
α,β,e, –d, –d; γ,d, +e–d, +e–d;
.
Otherwise, if we start with transformation (.) withx=y=z,
HA(α,β,β;γ,β;x,x,x) = ( –x)–α–βF
α,β; γ;
x
(–x)
,
we have another transformation
F()
e::β; –;α; –; –; –; α+β+d:: –; –; –;γ; –; –;
, ,
=
d–e,β+α+d d,β+α+d–e
F
α,β,e, –d; γ,+e–d, +e–d;
–
.
If we start with the identity betweenHBandF[, p., Eq. (.)] withx=y=z,
HB(α,β,β;γ,β,β;x,x,x) = ( –x)–α–βF
α,β;γ,β; x ( –x),
x ( –x)
,
we have
F()
e::β;β;α: –; –; –; α+β+d:: –; –; – :γ;β;β;
, ,
=
d–e,β+α+d d,β+α+d–e
m,n≥
(e)m+n(α)m+n(β)m+n( –d)m(–)m()n
(+e–d)m+n( +e–d)m+n(γ)m(β)nm!n!
.
It is interesting that the pattern of the double series on the right-hand side of the above transformation is similar to Exton’s [] triple hypergeometric series.
If we start with the relation betweenHAandG[, Eq. (.)] withy→–xandz→x,
HA(a+a– ,b,b;a,b;x, –x,x) = ( –x)–aG
b,a; –a, –a; x –x,
x –x
we have the following transformation:
F()
e::b; –;a+a– : –; –; –;
d+a:: –; –; – :a; –; –; , –,
=
d–e,d+a d,d+a–e
m,n≥
(e)m+n(b)m(a)n( –a)n–m( –a)m–n
(–)m+n( +e–d) m+nm!n!
,
withebeing a nonpositive integer.
Remark Many other transformations for the multiple hypergeometric functions can be
found by applying the beta integral method, and, as examples, we just present these re-sults. Interested readers can obtain some other transformation by using the beta integral method.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to this paper. They read and approved the final paper.
Author details
1Department of Information Technology, Hainan Medical College, Haikou, 571199, P.R. China.2Department of Mathematics, Shanghai University, Shanghai, 200444, P.R. China.
Acknowledgements
This work is supported by the National Natural Science Foundation of China (11201291 and 11301120), the Natural Science Foundation of Shanghai (12ZR1443800) and a grant of ‘The First-class Discipline of Universities in Shanghai’.
Received: 29 September 2013 Accepted: 12 November 2013 Published:12 Dec 2013
References
1. Rainville, ED: Special Functions. Macmillan Co., New York (1960). Reprinted by Chelsea, New York (1971)
2. Bailey, WN: Generalized Hypergeometric Series. Cambridge Tracts in Mathematics and Mathematical Physics, vol. 32. Cambridge University Press, Cambridge (1935). Reprinted by Stechert-Hafner, New York (1964)
3. Burchnall, JL, Chaundy, TW: Expansions of Appell’s double hypergeometric functions. II. Q. J. Math.12(1), 112-128 (1941)
4. Srivastava, HM, Panda, R: An integral representation for the product of two Jacobi polynomials. J. Lond. Math. Soc.
12(2), 419-425 (1976)
5. Buschman, RG, Srivastava, HM: Some identities and reducibility of Kampé de Fériet functions. Math. Proc. Camb. Philos. Soc.91, 435-440 (1982)
6. Exton, H: On the reducibility of the Kampé de Fériet function. J. Comput. Appl. Math.83, 119-121 (1997)
7. Karlsson, PW: Some reduction formulae for power series and Kampé de Fériet function. Indag. Math.87, 31-36 (1984) 8. Krupnikov, ED: A Register of Computer-Oriented Reduction Identities for the Kampé de Fériet Function. Novosibirsk
(1996)
9. Vid ¯unas, R: A generalization of Kummer identity. Rocky Mt. J. Math.32(2), 919-936 (2002)
10. Srivastava, HM, Karlsson, PW: Multiple Gaussian Hypergeometric Series. Horwood, Chichester (1985)
11. Krattenthaler, C, Srinivasa Rao, K: Automatic generation of hypergeometric identities by the beta integral method. J. Comput. Appl. Math.160, 159-173 (2003)
12. Choi, J, Rathie, AK, Srivastava, HM: Certain hypergeometric identities deducible by using the beta integral method. Bull. Korean Math. Soc.50, 1673-1681 (2013)
13. Erdélyi, A et al. (eds.) Higher Transcendental Functions, vol. 1. McGraw-Hill, New York (1953)
14. Manako, VV: A connection formula between double hypergeometric series2and3. Integral Transforms Spec. Funct.23(7), 503-508 (2012)
15. Slater, LJ: Generalized Hypergeometric Functions. Cambridge University Press, Cambridge (1966)
16. Vid ¯unas, R: Specialization of Appell’s functions to univariate hypergeometric functions. J. Math. Anal. Appl.355, 145-163 (2009)
17. Appell, P: Sur les fonctions hypergéométriques de plusieurs variables. Mémoir. Sci. Math. Gauthier-Villars, Paris (1925) 18. Appell, P, Kampé de Fériet, J: Fonctions hypergéométriques et hypersphériques Polynomes d’Hermite.
Gauthier-Villars, Paris (1926)
19. Exton, H: Hypergeometric functions of three variables. J. Indian Acad. Math.4(2), 113-119 (1982)
21. Erdélyi, A: Hypergeometric functions of two variables. Acta Math.83, 131-164 (1950)
22. Brychkov, YA: Handbook of Special Functions: Derivatives, Integrals, Series and Other Formulas. CRC Press, Boca Raton (2008)
23. Gradshteyn, IS, Ryzhyk, IM: Table of Integrals, Series, and Products. Academic Press, New York (2007)
24. Prudnikov, AP, Brychkov, YA, Marichev, OI: Integrals and Series, More Special Functions, vol. 3. Gordon & Breach, New York (1990)
25. Exton, H: On Srivastava’s symmetrical triple hypergeometric functionHB. J. Indian Acad. Math.25, 17-22 (2003)
26. Srivastava, HM: Hypergeometric functions of three variables. Ganita15, 97-108 (1964) 27. Srivastava, HM, Monacha, HL: A Treatise on Generating Functions. Horwood, Chichester (1984)
28. Srivastava, HM: Some integrals representing triple hypergeometric series. Rend. Circ. Mat. Palermo16(1), 99-115 (1967)
10.1186/1687-1847-2013-360
