ISSN2320-2882
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Paper Title
INTEGRAL REPRESENTATIONS AND GENERATING FUNCTIONS OF QUADRUPLE HYPERGEOMETRIC POLYNOMIALS
Authors
Dr. Pankaj Kumar Labh
Keywords
quadruple hypergeometric polynomials; multiple hypergeometric functions; Euler-type integrals; Laplace-type integrals; Pochhammer symbol; Exton-Srivastava functions; multiple Beta integrals; multivariate special functions
Abstract
Quadruple hypergeometric polynomials constitute a natural higher-dimensional extension of classical hypergeometric polynomials and arise in multivariate approximation theory, multiple orthogonality, and the analysis of systems of partial differential equations. Recent work on hypergeometric functions of several variables-especially those of Srivastava, Exton, and their generalizations-has established a variety of series, transformation, and integral representations for triple and quadruple hypergeometric functions. However, systematic integral representations tailored specifically to polynomial families of quadruple hypergeometric type remain comparatively underdeveloped.
In this paper we introduce a class of quadruple hypergeometric polynomials M_n (x_1,x_2,x_3,x_4) arising from a truncated quadruple hypergeometric series, and we derive several integral representations of Euler- and Laplace-type. Our approach combines classical techniques based on Beta and Gamma integrals with operational methods and factorization of the Pochhammer symbol, extending ideas used earlier for triple hypergeometric functions and Exton-Srivastava quadruple functions. The resulting integral formulas provide new analytical tools for studying convergence, asymptotic behavior, and structural properties (such as orthogonality and generating functions) of these polynomials. In addition, we discuss numerical implications and outline how the derived representations can be exploited for efficient computation via multidimensional quadrature and Monte Carlo methods.
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IJCRT — journal style
"INTEGRAL REPRESENTATIONS AND GENERATING FUNCTIONS OF QUADRUPLE HYPERGEOMETRIC POLYNOMIALS", IJCRT - International Journal of Creative Research Thoughts (www.ijcrt.org), ISSN:2320-2882, Vol.13, Issue 2, page no.i730-i740, February 2025, Available :https://ijcrt.org/papers/IJCRT25A2027.pdf
APA — 7th edition
Labh, P. K. (2025). INTEGRAL REPRESENTATIONS AND GENERATING FUNCTIONS OF QUADRUPLE HYPERGEOMETRIC POLYNOMIALS. International Journal of Creative Research Thoughts, 13(2), i730-i740. https://ijcrt.org/viewpaperforall.php?paper=IJCRT25A2027
MLA — 9th edition
Labh, Pankaj Kumar. "INTEGRAL REPRESENTATIONS AND GENERATING FUNCTIONS OF QUADRUPLE HYPERGEOMETRIC POLYNOMIALS." International Journal of Creative Research Thoughts, vol. 13, no. 2, 2025, pp. i730-i740, https://ijcrt.org/viewpaperforall.php?paper=IJCRT25A2027.
Chicago — 17th, bibliography
Labh, Pankaj Kumar. "INTEGRAL REPRESENTATIONS AND GENERATING FUNCTIONS OF QUADRUPLE HYPERGEOMETRIC POLYNOMIALS." International Journal of Creative Research Thoughts 13, no. 2 (2025): i730-i740. https://ijcrt.org/viewpaperforall.php?paper=IJCRT25A2027.
Harvard — author–date
Labh, P.K. (2025) 'INTEGRAL REPRESENTATIONS AND GENERATING FUNCTIONS OF QUADRUPLE HYPERGEOMETRIC POLYNOMIALS', International Journal of Creative Research Thoughts, 13(2), pp. i730-i740. Available at: https://ijcrt.org/viewpaperforall.php?paper=IJCRT25A2027
IEEE — numbered reference
P. K. Labh, "INTEGRAL REPRESENTATIONS AND GENERATING FUNCTIONS OF QUADRUPLE HYPERGEOMETRIC POLYNOMIALS," IJCRT, vol. 13, no. 2, pp. i730-i740, Feb. 2025.
Vancouver — biomedical
Labh PK. INTEGRAL REPRESENTATIONS AND GENERATING FUNCTIONS OF QUADRUPLE HYPERGEOMETRIC POLYNOMIALS. IJCRT. 2025 Feb;13(2):i730-i740.
AMA — 11th edition
Labh PK. INTEGRAL REPRESENTATIONS AND GENERATING FUNCTIONS OF QUADRUPLE HYPERGEOMETRIC POLYNOMIALS. IJCRT. 2025;13(2):i730-i740. Accessed at https://ijcrt.org/viewpaperforall.php?paper=IJCRT25A2027
BibTeX — Zotero, Mendeley, LaTeX
@article{IJCRT25A2027,
author = {Pankaj Kumar Labh},
title = {INTEGRAL REPRESENTATIONS AND GENERATING FUNCTIONS OF QUADRUPLE HYPERGEOMETRIC POLYNOMIALS},
journal = {International Journal of Creative Research Thoughts},
volume = {13},
number = {2},
pages = {i730-i740},
year = {2025},
month = {February},
issn = {2320-2882},
url = {https://ijcrt.org/viewpaperforall.php?paper=IJCRT25A2027}
}
RIS — EndNote, RefWorks
TY - JOUR
AU - Labh, Pankaj Kumar
TI - INTEGRAL REPRESENTATIONS AND GENERATING FUNCTIONS OF QUADRUPLE HYPERGEOMETRIC POLYNOMIALS
T2 - International Journal of Creative Research Thoughts
JA - IJCRT
VL - 13
IS - 2
PY - 2025
SN - 2320-2882
UR - https://ijcrt.org/viewpaperforall.php?paper=IJCRT25A2027
SP - i730-i740
ER -
Issue
Volume 13 Issue 2,
February-2025
Pages : i730-i740
Article Information
- Published In
- International Journal of Creative Research Thoughts — Volume 13, Issue 2, February 2025
- Pages
- i730-i740
- Published
- February 2025
- Registration ID
- IJCRT_299875
- Published Paper ID
- IJCRT25A2027
- Research Area
- Science and Technology
- Author Type
- Indian Author
- Location
- India
- Peer Review
- Request Peer Review Report
- Licence
- CC BY 4.0 · Open Access
- ISSN & Impact Factor
- 2320-2882 · IF 9.17
- Downloads
- 183
- Permanent Link
- https://ijcrt.org/viewpaperforall.php?paper=IJCRT25A2027
- Published Paper PDF
- https://ijcrt.org/papers/IJCRT25A2027.pdf
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