Identity-type functions and polynomials
The authors study identity-type functions and polynomials. An identity-type function \(e_\varphi\) is defined by \[ e_\varphi\circ \varphi=\varphi. \] The authors solve a general convolution-type equation and deduce the results for identity-type function.The main results obtained are given in the following two theorems: Theorem 1. Let \(\varphi\) and \(y\) be given functions analytic at zero such that the series \(e(t): \sum^\infty_{n=0}\frac{(-1)^n\psi(n;0)t^n}{\varphi(n+1;0)(n!)^2}\), is uniformly convergent for \(t\geq 0\), then \((e\circ \varphi)(t)=\psi(t)\). Theorem 2. Let \(\varphi(t) := (1 + t)^{-c}\) \((0 < c < 1)\). Then the function \(e(t):=(c-1)F[c,-c;1;t]\), where \(F[c,-c;1;t]\) is the Gauss hypergeometric function, is the identity-type function for the above function \(\varphi\) having the representation given in the book [\textit{I. S. Gradshteyn} and \textit{R. V. Ryzhik}, Table of integrals, series, and products, San Diego, CA: Academic Press (2000; Zbl 0981.65001)]. The above results give rise to the definition of sequence of polynomials , which are solutions of second-order linear differential equations. The associated identity-type polynomials are defined by \(e_n(t):= F[n,-n;1;t]\) \((n = 1,2,3,\dots)\). Orthogonality relations for these polynomials are obtained, as \[ \int^1_0 e_n(t)e_m(t)\frac{dt}{1-t}=0\quad (\forall m\neq n; n,m>1)\;\text{ and }\;\int^1_0 e^2_n(t) \frac{dt}{1-t}= \frac1{2n}\quad \forall (n=1,2,3,\dots). \] Finally, the relevance for solution of boundary value problems by a series and convergence of the series are briefly discussed.
scientific article; zbMATH DE number 5282481