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The method employed to deduce the latter is also im- plemented to obtain a Kummer-type transformation formula for r+1Fr+1(x) that was recently derived in a different way. Mathematics Subject Classification: 33C15, 33C20, 33C50. Keywords: Generalized hypergeometric function, Euler transformation, Kummer trans-.
21 Jan 2011 Classical fractional-linear and quadratic transformations are due to Euler, Pfaff, Gauss and Kummer. In. [Gou81] Goursat gave a list of transformations of degree 3, 4 and 6. It has been widely assumed that there are no other algebraic transformations, unless hypergeometric functions are algebraic functions.
Hypergeometric Functions. Reading. Problems. Introduction. The hypergeometric function F(a, b; c; x) is defined as. F(a; b; c; x) = 2F1(a, b; c; x) = F(b, a; c; x). = 1+ ab c x + a(a + 1)b(b + 1) c(c + 1) x2. 2! + ···. = o. ? n="0". (a)n(b)n. (c)n xn n! |x| < 1, c = 0, ?1, ?2, where. 2 ? refers to number of parameters in numerator.
1F1 and 2F0 (equivalently, the Kummer U-function) and the Gauss hypergeometric func- A function f(z) = ?. ? k="0" c(k)zk is called hypergeometric if the Taylor coefficients c(k) form a hypergeometric sequence, meaning that they satisfy a first-order recurrence In general, the Kummer transformation 1F1(a, b, z) = ez.
quadratic transformations for the former function, as well as a summation theorem when x = 1, are also considered. Mathematics Subject Classification: 33C15, 33C20. Keywords: Generalized hypergeometric function, Euler transformation, Kummer trans- formation, Quadratic transformations, Summation theorem, Zeros of
We also have a Barnes-type integral representation for the confluent hypergeometric function. In order to find this representation we compute its Mellin transform. By using Kummer's transformation formula (4) we obtain. ? ?. 0 zs?1. 1F1. (a c. ; -z. ) dz = ? ?. 0 zs?1e. ?z. 1F1. (c - a c. ; z. ) dz. = ?. ? n="0". (c - a)n. (c)n n!
H. ExtonOn the reducibility of the Kampe de Feriet function. J. Comput. Appl. Math., 83 (1997), pp. 119-121. [2]. A.R. MillerOn a Kummer-type transformation for the generalized hypergeometric function 2 F 2. J. Comput. Appl. Math., 157 (2003), pp. 507-509. [3]. R.B. Paris, D. KaminskiAsymptotics and Mellin–Barnes Integrals.
1 Feb 2008 In general, the 24 hypergeometric series represent 6 different Gauss hypergeometric functions, since Euler's and Pfaff's formulas [AAR99, Theorem 2.2.5] identify four Kummer's series with each other. We refer to those identities as the Euler-. Pfaff transformations. If we consider permutation of the upper
16.1 KUMMER FUNCTIONS. 321 are known as Kummer transformations [AS64, p. 505]. There are several connections between the confluent hypergeometric func- tions and the elementary functions as well as the error function, the loga- rithmic integral and functions related to the gamma function. Example 16.1.1: 1F1(1; a
25 Jan 2015 mation formulas contiguous to that of Kummer's second transformation for the confluent hypergeometric function 1F1 using a differential equation approach. Mathematics Subject Classification: 33C20. Keywords: Confluent hypergeometric function, Kummer's second theorem, hypergeometric differential
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