416 research outputs found
Euler-type transformations for the generalized hypergeometric function r+2Fr+1(x)
We provide generalizations of two of Euler’s classical transformation formulas for the Gauss hypergeometric function extended to the case of the generalized hypergeometric function r+2 F r+1(x) when there are additional numeratorial and denominatorial parameters differing by unity. The method employed to deduce the latter is also implemented to obtain a Kummer-type transformation formula for r+1 F r+1 (x) that was recently derived in a different way
Transformation formulas for the generalized hypergeometric function with integral parameter differences
Transformation formulas of Euler and Kummer-type are derived respectively for the generalized hypergeometric functions r+2Fr+1(x) and r+1Fr+1(x), where r pairs of numeratorial and denominatorial parameters differ by positive integers. Certain quadratic transformations for the former function, as well as a summation theorem when x = 1, are also considered.<br/
Clausen's series 3F2(1) with integral parameter differences and transformations of the hypergeometric function 2F2(x)
We obtain summation formulas for the hypergeometric series 3 F 2(1) with at least one pair of numeratorial and denominatorial parameters differing by a negative integer. The results derived for the latter are used to obtain Kummer-type transformations for the generalized hypergeometric function 2 F 2(x) and reduction formulas for certain Kampé de Fériet functions. Certain summations for the partial sums of the Gauss hypergeometric series 2 F 1(1) are also obtained
Certain transformations and summations for generalized hypergeometric series with integral parameter differences
Certain transformation and summation formulas for generalized hypergeometric series with integral parameter differences are derived
Recursion Rules for the Hypergeometric Zeta Functions
The hypergeometric zeta function is defined in terms of the zeros of the
Kummer function M(a, a + b; z). It is established that this function is an
entire function of order 1. The classical factorization theorem of Hadamard
gives an expression as an infinite product. This provides linear and quadratic
recurrences for the hypergeometric zeta function. A family of associated
polynomials is characterized as Appell polynomials and the underlying
distribution is given explicitly in terms of the zeros of the associated
hypergeometric function. These properties are also given a probabilistic
interpretation in the framework of Beta distributions
Some properties of generalized hypergeometric Appell polynomials
In this paper, we present a new real-valued Appell-type polynomial family
every member of which is expressed by mean of a generalized
hypergeometric function. The generating exponential function of this type of
polynomials is firstly discovered and the proof that they are of Appell-type
ones is given. We present the differential operator formal power series
representation as well as an explicit formula over the standard basis, and
establish a new identity for the generalized hypergeometric function. Besides,
we derive the addition, the multiplication and some other formulas for this
polynomial famil
An alternative proof of the extended SaalschĂĽtz summation theorem for the <sub>r + 3</sub>F<sub>r + 2</sub>(1) series with applications
A simple proof is given of a new summation formula recently added in the literature for a terminating r + 3Fr + 2(1) hypergeometric series for the case when r pairs of numeratorial and denominatorial parameters differ by positive integers. This formula represents an extension of the well-known Saalschütz summation formula for a 3F2(1) series. Two applications of this extended summation formula are discussed. The first application extends two identities given by Ramanujan and the second, which also employs a similar extension of the Vandermonde–Chu summation theorem for the 2F1 series, extends certain reduction formulas for the Kampé de Fériet function of two variables given by Exton and Cvijović & Miller
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