# Relations between Lauricella’s triple hypergeometric function ${F}_{A}^{(3)}(x,y,z)$ and Exton’s function ${X}_{8}$

- Junesang Choi
^{1}Email author and - Arjun K Rathie
^{2}

**2013**:34

https://doi.org/10.1186/1687-1847-2013-34

© Choi and Rathie; licensee Springer. 2013

**Received: **14 December 2012

**Accepted: **21 January 2013

**Published: **14 February 2013

## Abstract

Very recently Choi *et al.* derived some interesting relations between Lauricella’s triple hypergeometric function ${F}_{A}^{(3)}(x,y,z)$ and the Srivastava function ${F}^{(3)}[x,y,z]$ by simply splitting Lauricella’s triple hypergeometric function ${F}_{A}^{(3)}(x,y,z)$ into eight parts. Here, in this paper, we aim at establishing eleven new and interesting transformations between Lauricella’s triple hypergeometric function ${F}_{A}^{(3)}(x,y,z)$ and Exton’s function ${X}_{8}$ in the form of a single result. Our results presented here are derived with the help of two general summation formulae for the terminating ${}_{2}F_{1}(2)$ series which were very recently obtained by Kim *et al.* and also include the relationship between ${F}_{A}^{(3)}(x,y,z)$ and ${X}_{8}$ due to Exton.

**MSC:** 33C20, 44A45.

## Keywords

## 1 Introduction and preliminaries

*complex*numbers. For

*generalized hypergeometric series*${}_{p}F_{q}$ with

*p*numerator parameters ${\alpha}_{1},\dots ,{\alpha}_{p}$ and

*q*denominator parameters ${\beta}_{1},\dots ,{\beta}_{q}$ is defined by (see, for example, [[1], Chapter 4]; see also [[2], p.73]):

*shifted factorial*since

it being *understood conventionally* that ${(0)}_{0}:=1$.

Motivated essentially by the works by Lardner [7] and Carlson [8], by simply splitting Lauricella’s triple hypergeometric function ${F}_{A}^{(3)}(x,y,z)$ into eight parts, very recently Choi *et al.* [5] presented several relationships between ${F}_{A}^{(3)}(x,y,z)$ and the Srivastava function ${F}^{(3)}[x,y,z]$ (see also [9]). The widely-investigated Srivastava’s triple hypergeometric function ${F}^{(3)}[x,y,z]$, which was introduced over four decades ago by Srivastava [[10], p.428] (see also [5], [[3], p.44, Equation 1.5(14)] and [[6], p.69, Equation 1.7(39)]), provides an interesting unification (and generalization) of Lauricella’s 14 triple hypergeometric functions ${F}_{1},\dots ,{F}_{14}$ (see [11], [[12], pp.113-114]) and Srivastava’s three functions ${H}_{A}$, ${H}_{B}$ and ${H}_{C}$ (see [[13], pp.99-100]; see also [14–16], [[3], p.43] and [[6], pp.60-68]).

In fact, in 1982 Exton [17] published a very interesting and useful research paper in which he encountered a number of triple hypergeometric functions of second order whose series representations involve such products as ${(a)}_{2m+2n+p}$ and ${(a)}_{2m+n+p}$ and introduced a set of 20 distinct triple hypergeometric functions ${X}_{1}$ to ${X}_{20}$ and also gave their integral representations of Laplacian type which include the confluent hypergeometric functions ${}_{0}F_{1}$, ${}_{1}F_{1}$, a Humbert function ${\psi}_{1}$ and a Humbert function ${\varphi}_{2}$ in their kernels. It is not out of place to mention here that Exton’s functions ${X}_{1}$ to ${X}_{20}$ have been studied a lot until today; see, for example, the works [12, 18–23] and [24]. Moreover, Exton [17] presented a large number of very interesting transformation formulas and reducible cases with the help of two known results which are called in the literature Kummer’s first and second transformations or theorems.

Here, in this paper, we aim at establishing eleven new and interesting transformations between Lauricella’s triple hypergeometric function ${F}_{A}^{(3)}(x,y,z)$ and Exton’s function ${X}_{8}$ in the form of a single result. Our results presented here are derived with the help of two general summation formulae for the terminating ${}_{2}F_{1}(2)$ series which were very recently obtained by Kim *et al.* [25] and also include the relationship between ${F}_{A}^{(3)}(x,y,z)$ and ${X}_{8}$ due to Exton [17].

## 2 Results required

*x*and its modulus is denoted by $|x|$, and the coefficients ${\mathcal{A}}_{j}$ and ${\mathcal{B}}_{j}$ are given in Table 1.

**Contiguous relation coefficients**

j | ${\mathcal{A}}_{\mathbf{j}}$ | ${\mathcal{B}}_{\mathbf{j}}$ |
---|---|---|

5 | $-4{(1-\alpha -2n)}^{2}+2(1-\alpha )(1-\alpha -2n)+$ ${(1-\alpha )}^{2}+22(1-\alpha -2n)-13(1-\alpha )-20$ | $4{(\alpha +2n)}^{2}-2(1-\alpha )(\alpha +2n)+$ ${(1-\alpha )}^{2}+34(\alpha +2n)+(1-\alpha )+62$ |

4 | 2( | 4( |

3 | − | −3 |

2 | − | −2 |

1 | −1 | 1 |

0 | 1 | 0 |

−1 | 1 | 1 |

−2 | 1 − | 2 |

−3 | 1 − | 3 − 3 |

−4 | 2(1 − 2 | 4(1 − |

−5 | $4{(1-\alpha -2n)}^{2}-2(1-\alpha )(1-\alpha -2n)$ $-{(1-\alpha )}^{2}+8(1-\alpha -2n)+7\alpha -7$ | $4{(\alpha +2n)}^{2}+2(1-\alpha )(\alpha +2n)-$ ${(1-\alpha )}^{2}-16(\alpha +2n)+\alpha -1$ |

## 3 Main transformation formulae

where the coefficients ${\mathcal{C}}_{j}$ and ${\mathcal{D}}_{j}$ can be obtained by simply changing *n* and *α* into *r* and ${c}_{1}$, respectively, in Table 1 of ${\mathcal{A}}_{j}$ and ${\mathcal{B}}_{j}$.

*Proof*For convenience and simplicity, by denoting the left-hand side of (3.1) by

*S*and using the series definition of ${F}_{A}^{(3)}(x,y,z)$ as given in (1.4), after a little simplification, we have

*r*into even and odd integers, we have

Finally, using the known results (2.1) and (2.2), after a little simplification, we easily arrive at the right-hand side of (3.1). This completes the proof of (3.1). □

## 4 Special cases

In our main formula (3.1), if we take $j=0,\pm 1$ and ±2, after a little simplification, and interpret the respective resulting right-hand sides with the definition of Exton’s triple hypergeometric series ${X}_{8}$ given in (1.5), we get the following very interesting relations between ${F}_{A}^{(3)}(x,y,z)$ and ${X}_{8}$:

**Remark** Clearly, Equation (4.1) is Exton’s result (see [17]) and Equations (4.2) to (4.5) are closely related to it. The other special cases of (3.1) can also be expressed in terms of ${X}_{8}$ in a similar manner.

## Declarations

### Acknowledgements

Dedicated to Professor Hari M Srivastava.

This paper was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (2012-0002957).

## Authors’ Affiliations

## References

- Erdélyi A, Magnus W, Oberhettinger F, Tricomi FG
**I.**In*Higher Transcendental Functions*. McGraw-Hill, New York; 1953.Google Scholar - Rainville ED:
*Special Functions*. Macmillan Co., New York; 1960. Reprinted by Chelsea, New York (1971)MATHGoogle Scholar - Srivastava HM, Karlsson PW:
*Multiple Gaussian Hypergeometric Series*. Horwood, Chichester; 1985.MATHGoogle Scholar - Appell P, Kampé de Fériet J:
*Fonctions Hypergeometriques et Hyperspheriques; Polynomes d’Hermite*. Gauthier-Villars, Paris; 1926.MATHGoogle Scholar - Choi J, Hasanov A, Srivastava HM:Relations between Lauricella’s triple hypergeometric function ${F}_{A}^{(3)}(x,y,z)$ and the Srivastava function ${F}^{(3)}[x,y,z]$.
*Integral Transforms Spec. Funct.*2012, 23(1):69-82. 10.1080/10652469.2011.596710MATHMathSciNetView ArticleGoogle Scholar - Srivastava HM, Manocha HL:
*A Treatise on Generating Functions*. Horwood, Chichester; 1984.MATHGoogle Scholar - Lardner TJ:Relations between ${}_{0}F_{3}$
**and Bessel functions.***SIAM Rev.*1969, 11: 69-72. 10.1137/1011007MATHMathSciNetView ArticleGoogle Scholar - Carlson BC:Some extensions of Lardner’s relations between ${}_{0}F_{3}$
**and Bessel functions.***SIAM J. Math. Anal.*1970, 1(2):232-242. 10.1137/0501021MATHMathSciNetView ArticleGoogle Scholar - Choi J, Kim YS, Hasanov A:Relations between the hypergeometric function of Appell ${F}_{3}$
**and Kampé de Fériet functions.***Miskolc Math. Notes*2011, 12(2):131-148.MATHMathSciNetGoogle Scholar - Srivastava HM:
**Generalized Neumann expansions involving hypergeometric functions.***Proc. Camb. Philos. Soc.*1967, 63: 425-429. 10.1017/S0305004100041359MATHView ArticleGoogle Scholar - Lauricella G:
**Sulle funzioni ipergeometriche a più variabili.***Rend. Circ. Mat. Palermo*1893, 7: 111-158. 10.1007/BF03012437MATHView ArticleGoogle Scholar - Kim YS, Choi J, Rathie AK:Remark on two results by Padmanabham for Exton’s triple hypergeometric series ${X}_{8}$.
*Honam Math. J.*2005, 27(4):603-608.Google Scholar - Srivastava HM:
**Hypergeometric functions of three variables.***Ganita*1964, 15(2):97-108.MATHMathSciNetGoogle Scholar - Choi J, Hasanov A, Srivastava HM, Turaev M:
**Integral representations for Srivastava’s triple hypergeometric functions.***Taiwan. J. Math.*2011, 15: 2751-2762.MATHMathSciNetGoogle Scholar - Srivastava HM:
**Some integrals representing triple hypergeometric functions.***Rend. Circ. Mat. Palermo*1967, 16: 99-115. 10.1007/BF02844089MATHMathSciNetView ArticleGoogle Scholar - Turaev M:Decomposition formulas for Srivastava’s hypergeometric function ${H}_{A}$
**on Saran functions.***J. Comput. Appl. Math.*2009, 233: 842-846. 10.1016/j.cam.2009.02.050MATHMathSciNetView ArticleGoogle Scholar - Exton H:
**Hypergeometric functions of three variables.***J. Indian Acad. Math.*1982, 4: 113-119.MATHMathSciNetGoogle Scholar - Choi J, Hasanov A, Turaev M:
**Decomposition formulas and integral representations for some Exton hypergeometric functions.***J. Chungcheong Math. Soc.*2011, 24(4):745-758.Google Scholar - Choi J, Hasanov A, Turaev M:Linearly independent solutions for the hypergeometric Exton functions ${X}_{1}$ and ${X}_{2}$.
*Honam Math. J.*2010, 32(2):223-229.Google Scholar - Choi J, Hasanov A, Turaev M:Certain integral representations of Euler type for the Exton function ${X}_{5}$.
*Honam Math. J.*2010, 32(3):389-397. 10.5831/HMJ.2010.32.3.389MATHMathSciNetView ArticleGoogle Scholar - Choi J, Hasanov A, Turaev M:Certain integral representations of Euler type for the Exton function ${X}_{2}$.
*J. Korean Soc. Math. Edu., Ser. B, Pure Appl. Math.*2010, 17(4):347-354.MATHMathSciNetGoogle Scholar - Kim YS, Rathie AK:On an extension formulas for the triple hypergeometric ${X}_{8}$
**due to Exton.***Bull. Korean Math. Soc.*2007, 44(4):743-751. 10.4134/BKMS.2007.44.4.743MATHMathSciNetView ArticleGoogle Scholar - Kim YS, Rathie AK, Choi J:Another method for Padmanabham’s transformation formula for Exton’s triple hypergeometric series ${X}_{8}$.
*Commun. Korean Math. Soc.*2009, 24(4):517-521. 10.4134/CKMS.2009.24.4.517MATHMathSciNetView ArticleGoogle Scholar - Lee SW, Kim YS:
**An extension of the triple hypergeometric series by Exton.***Honam Math. J.*2009, 31(1):61-71.Google Scholar - Kim YS, Rakha MA, Rathie AK:
**Generalization of Kummer’s second summation theorem with applications.***Comput. Math. Math. Phys.*2010, 50(3):387-402. 10.1134/S0965542510030024MathSciNetView ArticleGoogle Scholar - Choi J:
**Notes on formal manipulations of double series.***Commun. Korean Math. Soc.*2003, 18(4):781-789.MATHMathSciNetView ArticleGoogle Scholar

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