Confluent functions, Laguerre polynomials and their (generalized) bilinear integrals

TL;DR

The paper analyzes confluent hypergeometric theory through the ${}_1F_1$ and ${}_2F_0$ formalisms and their eigenfunctions, focusing on Tricomi's functions $U_{\theta,\alpha}$ and their bilinear integrals with weight $e^{-z}z^{\alpha}$. It introduces a Lie-algebraic parameterization to unify the confluent equations, derives explicit connection formulas, and studies degenerate and Bessel-reduction limits, including the generalized integral that extends bilinear forms to all $\alpha$, even in anomalous (integer) cases. Laguerre polynomials are treated as special cases, yielding generalized Gram matrices that encode (pseudo-)orthogonality beyond the classical parameter range and linking back to Tricomi/bilinear structures. The results provide closed-form bilinear integrals, generating function representations, and regularization schemes that are relevant for applications ranging from orthogonal polynomial theory to renormalization-inspired models in mathematical physics.

Abstract

We review properties of confluent functions and the closely related Laguerre polynomials, and determine their bilinear integrals. As is well-known, these integrals are convergent only for a limited range of parameters. However, when one uses the generalized integral they can be computed essentially without restricting the parameters. This gives the (generalized) Gram matrix of Laguerre polynomials. If the parameters are not negative integers, then Laguerre polynomials are orthogonal, or at least pseudo-orthogonal in the case of generalized integrals. For negative integer parameters, the orthogonality relations are more complicated.

Theorems & Definitions (7)

• proof
• proof
• proof
• proof : Alternative proof of Thm. \ref{['prop:anomalous_laguerre']}
• proof
• proof
• proof