Four Families of Summation Formulas for 4F3(1) with Application

Author:

Kumar Belakavadi Radhakrishna Srivatsa1ORCID,Rathie Arjun K.2ORCID,Choi Junesang3ORCID

Affiliation:

1. Department of Mathematics, Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal 576104, India

2. Department of Mathematics, Vedant College of Engineering and Technology, Bundi 323001, India

3. Department of Mathematics, Dongguk University, Gyeongju 38066, Republic of Korea

Abstract

A collection of functions organized according to their indexing based on non-negative integers is grouped by the common factor of fixed integer N. This grouping results in a summation of N series, each consisting of functions partitioned according to this modulo N rule. Notably, when N is equal to two, the functions in the series are divided into two subseries: one containing even-indexed functions and the other containing odd-indexed functions. This partitioning technique is widely utilized in the mathematical literature and finds applications in various contexts, such as in the theory of hypergeometric series. In this paper, we employ this partitioning technique to establish four distinct families of summation formulas for F34(1) hypergeometric series. Subsequently, we leverage these summation formulas to introduce eight categories of integral formulas. These integrals feature compositions of Beta function-type integrands and F23(x) hypergeometric functions. Additionally, we highlight that our primary summation formulas can be used to derive some well-known summation results.

Publisher

MDPI AG

Reference26 articles.

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2. Rainville, E.D. (1960). Special Functions, Macmillan Company. Reprinted by Chelsea Publishing Company: New York, NY, USA, 1971.

3. Bailey, W.N. (1935). Generalized Hypergeometric Series, Cambridge University Press. Reprinted by Stechert-Hafner, New York, NY, USA, 1964.

4. Slater, L.J. (1966). Generalized Hypergeometric Functions, Cambridge University Press.

5. Srivastava, H.M., and Choi, J. (2012). Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers.

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