# Index transforms with the product of the associated Legendre functions

## TL;DR

This paper introduces new index transforms involving the product of associated Legendre functions, explores their properties in Lebesgue spaces, proves inversion formulas, and applies these results to solve a boundary value problem for a third order PDE.

## Contribution

It presents novel index transforms with associated Legendre functions, analyzes their properties, and demonstrates their application to boundary value problems.

## Key findings

- Established mapping properties in Lebesgue spaces
- Derived inversion formulas for the transforms
- Applied results to solve a boundary value problem

## Abstract

Index transforms with the product of the associated Legendre functions are introduced. Mapping properties are investigated in the Lebesgue spaces. Inversion formulas are proved. The results are applied to solve a boundary value problem in a wedge for a third order partial differential equation.