A UNIFIED CLASS OF POLYNOMIALS

Hukum Chand Agrawal

Abstract: In this paper we propose to study the polynomial set $\left\{f^{(\alpha)}_n\right\}(x)$ satisfying the functional relation $$ T(\Delta_\alpha)\left\{f^{(\alpha)}_n(x)\right\}= f^{(\alpha+1)}_{n-1}(x), \qquad n=1,2,3,\dots, $$ where $f(\alpha)_n(x)$ is the polynomial of degree $n$ in $x$ and $T$ is the operator of infinite order defined by $$ T(\Delta_\alpha)= \sum_{k=0}^\infty h_k^{(\alpha)}\Delta_\alpha^{k+1}, \enskip h_0^{(\alpha)}\neq 0, $$ in which $\Delta_\alpha \{f(\alpha)\}= f(\alpha+1)-f(\alpha)$.

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