Factorization-driven operator framework for Delta_(delta) Gould–Hopper Appell polynomial families and their zeros via graphical representation

Shivtej Annaso Patil; Waseem Ahmad Khan; Wael Mahmoud Mohammad Salameh; Prakash Jadhav; Shahid Ahmad Wani

doi:10.46481/jnsps.2026.3662

Authors

Shivtej Annaso Patil
Symbiosis Institute of Technology, Pune Campus, Symbiosis International (Deemed University), Pune, India;

General Engineering Department, DKTE Society’s Textile and Engineering Institute, Ichalkaranji 416115, Maharashtra, India
Waseem Ahmad Khan
Department of Electrical Engineering, Prince Mohammad Bin Fahd University, P.O. Box 1664, Al Khobar 31952, Saudi Arabia
Wael Mahmoud Mohammad Salameh
Faculty of Information Technology, Abu Dhabi University, Abu Dhabi, United Arab Emirates
Prakash Jadhav
Department of Mechanical Engineering, SRM University-AP, Amravati 522240, Andhra Pradesh, India
Shahid Ahmad Wani
[email protected]

Symbiosis Institute of Technology, Pune Campus, Symbiosis International (Deemed University), Pune, India

Keywords:

Δδ Gould–Hopper Appell polynomials, Recurrence relation, Shift operators, Difference equations, Complex zeros

Abstract

This article develops a factorization-based operator framework for two-variable discrete Gould--Hopper--Appell polynomials within Delta_(delta) -calculus. Following the Infeld--Hull factorization principle, the framework employs degree-dependent sequences of operators, rather than a single pair of ordinary quasi-monomial operators. From the generating function, we derive the recurrence relation, lowering and raising operators, and normalized integro-type operators. Their successive degree-matched compositions yield explicit difference, integro-difference, and partial-difference equations. Exact triangular Volterra integral representations are also established through the associated differential representation. The general results are specialized to Gould--Hopper--Bernoulli, Gould--Hopper--Euler, and Gould--Hopper--Genocchi families, with the Genocchi case treated separately. Numerical investigations of complex zeros, polynomial surfaces, generating-function reconstruction, and quantitative convergence as delta to 0 complement the analytical results and illustrate the discrete-to-continuous behavior of the proposed polynomial families.

Dimensions

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