Published Paper Details:

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.