Relations between Lauricella’s triple hypergeometric function and Exton’s function
© Choi and Rathie; licensee Springer. 2013
Received: 14 December 2012
Accepted: 21 January 2013
Published: 14 February 2013
Very recently Choi et al. derived some interesting relations between Lauricella’s triple hypergeometric function and the Srivastava function by simply splitting Lauricella’s triple hypergeometric function into eight parts. Here, in this paper, we aim at establishing eleven new and interesting transformations between Lauricella’s triple hypergeometric function and Exton’s function in the form of a single result. Our results presented here are derived with the help of two general summation formulae for the terminating series which were very recently obtained by Kim et al. and also include the relationship between and due to Exton.
MSC: 33C20, 44A45.
Keywordsgamma function hypergeometric functions of several variables multiple Gaussian hypergeometric series Exton’s triple hypergeometric series Gauss’s hypergeometric functions Lauricella’s triple hypergeometric functions
1 Introduction and preliminaries
it being understood conventionally that .
Motivated essentially by the works by Lardner  and Carlson , by simply splitting Lauricella’s triple hypergeometric function into eight parts, very recently Choi et al.  presented several relationships between and the Srivastava function (see also ). The widely-investigated Srivastava’s triple hypergeometric function , which was introduced over four decades ago by Srivastava [, p.428] (see also , [, p.44, Equation 1.5(14)] and [, p.69, Equation 1.7(39)]), provides an interesting unification (and generalization) of Lauricella’s 14 triple hypergeometric functions (see , [, pp.113-114]) and Srivastava’s three functions , and (see [, pp.99-100]; see also [14–16], [, p.43] and [, pp.60-68]).
In fact, in 1982 Exton  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 and and introduced a set of 20 distinct triple hypergeometric functions to and also gave their integral representations of Laplacian type which include the confluent hypergeometric functions , , a Humbert function and a Humbert function in their kernels. It is not out of place to mention here that Exton’s functions to have been studied a lot until today; see, for example, the works [12, 18–23] and . Moreover, Exton  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 and Exton’s function in the form of a single result. Our results presented here are derived with the help of two general summation formulae for the terminating series which were very recently obtained by Kim et al.  and also include the relationship between and due to Exton .
2 Results required
Contiguous relation coefficients
2(α + 1 + 2n)(α + 3 + 2n)−α(α + 3)
4(α + 2n + 3)
−α − 4n − 2
−3α − 4n − 6
−α − 1 − 2n
1 − α − 2n
1 − α − 4n
3 − 3α − 4n
2(1 − 2α − n)(3 − α − 2n)−(1 − α)(4 − α)
4(1 − α − 2n)
3 Main transformation formulae
where the coefficients and can be obtained by simply changing n and α into r and , respectively, in Table 1 of and .
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 and ±2, after a little simplification, and interpret the respective resulting right-hand sides with the definition of Exton’s triple hypergeometric series given in (1.5), we get the following very interesting relations between and :
Remark Clearly, Equation (4.1) is Exton’s result (see ) 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 in a similar manner.
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).
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