polyPn1

polyPn1 is a formal family of polynomials denoted P_n^1(x), defined for integer n ≥ 0 and real variable x with a fixed parameter equal to 1. In the polyPn1 framework, the sequence is presented as a generalization of classical orthogonal polynomials, designed to explore how a single parameter influences recurrence, zeros, and representation.

Construction and basic properties: The sequence starts with P_0^1(x) = 1 and P_1^1(x) = x. For n ≥ 1, P_{n+1}^1(x) is given by a three-term recurrence of the form P_{n+1}^1(x) = x P_n^1(x) − n P_{n−1}^1(x). Each P_n^1 is a polynomial of degree n in x, with coefficients determined solely by n and the fixed parameter 1. The recurrence implies a structured hierarchy among the polynomials and yields a compact, cumulative description via a generating function.

Generating function and representations: The polynomials are frequently encoded through a generating function G(t, x) = sum_{n≥0} P_n^1(x) t^n, which reflects the three-term recurrence and supports algebraic manipulations. In many treatments, P_n^1(x) admits a hypergeometric representation, linking polyPn1 to broader families of special functions and enabling closed-form or semi-analytic expressions for specific n.

Applications and context: PolyPn1 figures in studies of generalized orthogonal polynomials, spectral methods for differential equations, and moment problems where a single parameter modulates classical families. Its simple recurrence makes it a convenient test case for numerical schemes, quadrature construction, and symbolic manipulation, while offering a controlled bridge to more elaborate polynomial systems.

History and naming: The designation polyPn1 derives from the attempt to name a cohesive family that keeps a constant parameter 1 visible in its notation, with origins in contemporary explorations of polynomial generalizations initiated in the early 2020s.

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