Reduction formula of a double binomial sum

AbstractIn a recent paper, Miller derived a Kummer-type transformation for the generalised hypergeometric function pFp(x) when pairs of parameters differ by unity, by means of a reduction formula for a certain Kampé de Fériet function. An alternative and simpler derivation of this transformation is obtained here by application of the well-known Kummer transformation for the confluent hypergeometric function corresponding to p = 1.

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A NEW PROOF OF A REDUCTION FORMULA FOR THE APPELL SERIES F3 DUE TO BAILEY

Facta Universitatis Series Mathematics and Informatics ◽

10.22190/fumi1905849m ◽

2019 ◽

pp. 849

Author(s):

Gradimir V. Milovanović ◽

Arjun K. Rathie

Keyword(s):

Short Note ◽

Reduction Formula

In this short note, we provide a new proof of an interesting and useful reduction formula for the Appell series $F_{3}$ due to Bailey [{\it On the sum of a terminating ${}_3F_2(1)$}, Quart. J. Math. Oxford Ser. (2) {\bf4} (1953), 237--240].

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A REDUCTION FORMULA FOR NORMAL MULTIVARIATE INTEGRALS

Biometrika ◽

10.1093/biomet/41.3-4.351 ◽

1954 ◽

Vol 41 (3-4) ◽

pp. 351-360 ◽

Cited By ~ 136

Author(s):

R. L. PLACKETT

Keyword(s):

Reduction Formula

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The numerical evaluation of a class of integrals. II

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100031443 ◽

1956 ◽

Vol 52 (3) ◽

pp. 442-448 ◽

Cited By ~ 6

Author(s):

S. C. Das

Keyword(s):

Normal Distribution ◽

Infinite Series ◽

Correlation Matrix ◽

Multivariate Normal Distribution ◽

Numerical Evaluation ◽

Numerical Quadrature ◽

Multivariate Normal ◽

Reduction Formula ◽

Image Position ◽

Finite Sum

Consider the integralwhere x1, x2, …, xN are jointly distributed in a multivariate normal distribution f(x1, x2, …, xN) with (pij) as the correlation matrix. The integral has been expressed in an infinite series of tetrachoric functions for N≥2. The infinite series is not only complicated, but also is very slowly convergent and is consequently not of much practical use. Plackett (8) obtains a reduction formula for expressing normal integrals in four variates as a finite sum of single integrals of tabulated functions. These integrals have then to be evaluated by a rather awkward numerical quadrature.

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