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2013•The Pure and Applied MathematicsOpen access

APPELL'S FUNCTION F 1 AND EXTON'S TRIPLE HYPERGEOMETRIC FUNCTION X 9

Junesang Choi, Arjun K. Rathie

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Abstract

In the theory of hypergeometric functions of one or several variables, a remarkable amount of mathematicians's concern has been given to develop their transformation formulas and summation identities. Here we aim at presenting explicit expressions (in a single form) of the following weighted Appell's function $F_1$ : $$(1+2x)^{-a}(1+2z)^{-b}F_1\;\(c,\;a,\;b;\;2c+j;\;\frac{4x}{1+2x},\;\frac{4z}{1+2z}\)\;(j=0,\;{\pm}1,\;{\ldots},\;{\pm}5)$$ in terms of Exton's triple hypergeometric $X_9$ . The results are derived with the help of generalizations of Kummer's second theorem very recently provided by Kim et al. A large number of very interesting special cases including Exton's result are also given.

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In the theory of hypergeometric functions of one or several variables, a remarkable amount of mathematicians's concern has been given to develop their transformation formulas and summation identities. Here we aim at presenting explicit expressions (in a single form) of the following weighted Appell's function $F_1$ : $$(1+2x)^{-a}(1+2z)^{-b}F_1\;\(c,\;a,\;b;\;2c+j;\;\frac{4x}{1+2x},\;\frac{4z}{1+2z}\)\;(j=0,\;{\pm}1,\;{\ldots},\;{\pm}5)$$ in terms of Exton's triple hypergeometric $X_9$ . The results are derived with the help of generalizations of Kummer's second theorem very recently provided by Kim et al. A large number of very interesting special cases including Exton's result are also given.

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Available abstract

In the theory of hypergeometric functions of one or several variables, a remarkable amount of mathematicians's concern has been given to develop their transformation formulas and summation identities. Here we aim at presenting explicit expressions (in a single form) of the following weighted Appell's function $F_1$ : $$(1+2x)^{-a}(1+2z)^{-b}F_1\;\(c,\;a,\;b;\;2c+j;\;\frac{4x}{1+2x},\;\frac{4z}{1+2z}\)\;(j=0,\;{\pm}1,\;{\ldots},\;{\pm}5)$$ in terms of Exton's triple hypergeometric $X_9$ . The results are derived with the help of generalizations of Kummer's second theorem very recently provided by Kim et al. A large number of very interesting special cases including Exton's result are also given.

Key concepts: Hypergeometric function, Mathematics, Function (biology), Transformation (genetics), Hypergeometric distribution, Pure mathematics, Basic hypergeometric series, Generalized hypergeometric function

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APPELL'S FUNCTION F 1 AND EXTON'S TRIPLE HYPERGEOMETRIC FUNCTION X 9 — Research Paper | ScholarLens