# Equations of hypergeometric type in the degenerate case

## TL;DR

This paper thoroughly analyzes the solutions with logarithmic singularities of the three main hypergeometric equations in the degenerate case where a parameter is an integer.

## Contribution

It provides a detailed examination of the logarithmic solutions for ${}_2F_1$, ${}_1F_1$, and ${}_1F_0$ hypergeometric equations when parameters are degenerate.

## Key findings

- Characterization of logarithmic solutions in degenerate cases
- Explicit formulas for solutions with singularities
- Insights into the structure of hypergeometric functions in special cases

## Abstract

We consider the three most important equations of hypergeometric type, ${}_2F_1$, ${}_1F_1$ and ${}_1F_0$, in the so-called degenerate case. In this case one of the parameters, usually denoted $c$, is an integer and the standard basis of solutions consists of a hypergeometric-type function and a function with a logarithmic singularity. This article is devoted to a thorough analysis of the latter solution to all three equations.