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Some Terminating K-Balanced Clausen Hypergeometric Summation Theorems

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Abstract

The main aim of this paper is to establish some new hypergeometric summation theorems (14), (34) and the results (41), (42), (43) and (44) (not recorded earlier and numerically verified using Mathematica software) for k-balanced terminating Clausen series in terms of the ratio of the product of Pochhammer symbols, by using a relation between two terminating Clausen series, decomposition of the ratio of two Pochhammer symbols, Chu–Vandermonde summation theorem, algebraic properties of gamma functions and Pochhammer symbols, representation of a linear, quadratic, cubic, bi-quadratic polynomials in terms of Pochhammer symbols and series rearrangement technique.

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Acknowledgements

The authors are very thankful to the reviewer for carefully reading the manuscript and for giving valuable suggestions and comments which significantly improve the understandability of the revised version of the paper.

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Authors and Affiliations

  1. Department of Applied Sciences and Humanities, Faculty of Engineering and Technology, Jamia Millia Islamia (A Central University), New Delhi, 110025, India

    Mohammad Idris Qureshi & Shakir Hussain Malik

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  1. Mohammad Idris Qureshi
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Our article has two authors (Both are contributed equally).

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Correspondence to Shakir Hussain Malik.

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Appendix

Appendix

The roots I, R, S of the cubic equation \( Mx^3+Vx^2+Wx+Y=0,\) are calculated by using Wolfram Mathematica 9.0 Software or Cardan’s method (Cardon’s method). The values of I, R and S are given as follows:

  • $$\begin{aligned}{} & {} I=\frac{-V}{3M}\nonumber \\{} & {} \quad -\frac{2^\frac{1}{3}(-V^2+3MW)}{3M\left\{ -2V^3+9MVW-27M^2Y +\sqrt{4(-V^2+3MW)^3+(-2V^3+9MVW-27M^2Y)^2}~\right\} ^\frac{1}{3}}\nonumber \\{} & {} \quad +\frac{\left\{ -2V^3+9MVW-27M^2Y+\sqrt{4(-V^2+3MW)^3 +(-2V^3+9MVW-27M^2Y)^2}\right\} ^\frac{1}{3}}{3\times 2^\frac{1}{3}M}\nonumber \\ \end{aligned}$$
    (51)
  • $$\begin{aligned}{} & {} R=\frac{-V}{3M}\nonumber \\{} & {} \quad +\frac{(1+i\sqrt{3})(-V^2+3MW)}{3\times 2^\frac{2}{3}M\left\{ -2V^3+9MVW-27M^2Y +\sqrt{4(-V^2+3MW)^3+(-2V^3+9MVW-27M^2Y)^2}\right\} ^\frac{1}{3}}\nonumber \\{} & {} \qquad -\frac{(1-i\sqrt{3})\left\{ -2V^3+9MVW-27M^2Y+\sqrt{4(-V^2+3MW)^3 +(-2V^3+9MVW-27M^2Y)^2}\right\} ^\frac{1}{3}}{6\times 2^\frac{1}{3}M}\nonumber \\ \end{aligned}$$
    (52)
  • $$\begin{aligned}{} & {} S=\frac{-V}{3M}\nonumber \\{} & {} \quad +\frac{(1-i\sqrt{3})(-V^2+3MW)}{3\times 2^\frac{2}{3}M\left\{ -2V^3+9MVW-27M^2Y +\sqrt{4(-V^2+3MW)^3+(-2V^3+9MVW-27M^2Y)^2}\right\} ^\frac{1}{3}}\nonumber \\{} & {} \quad -\frac{(1+i\sqrt{3})\left\{ -2V^3+9MVW-27M^2Y+\sqrt{4(-V^2+3MW)^3 +(-2V^3+9MVW-27M^2Y)^2}\right\} ^\frac{1}{3}}{6\times 2^\frac{1}{3}M}\nonumber \\ \end{aligned}$$
    (53)

    Therefore,

    $$\begin{aligned} I+R+S=\frac{-V}{M} \end{aligned}$$
    (54)
    $$\begin{aligned} IR+RS+SI=\frac{W}{M} \end{aligned}$$
    (55)

    and

    $$\begin{aligned} IRS=\frac{-Y}{M} ~~or~ MIRS=-Y=-T(T+1)(T+2)U(U+1)(U+2) \end{aligned}$$
    (56)

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Qureshi, M.I., Malik, S.H. Some Terminating K-Balanced Clausen Hypergeometric Summation Theorems. Int. J. Appl. Comput. Math 9, 59 (2023). https://doi.org/10.1007/s40819-023-01536-0

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