# Construction of general forms of ordinary generating functions for more families of numbers and multiple variables polynomials

## TL;DR

This paper develops comprehensive generating functions for a wide variety of special numbers and polynomials, deriving new identities, recurrence relations, and formulas using advanced mathematical transforms.

## Contribution

It introduces a unified approach to construct generating functions for diverse polynomial families, including Fibonacci, Lucas, Chebyshev, and others, with new identities and formulas.

## Key findings

- New identities and relations derived via Euler transform and Lambert series.
- Recurrence relations established using differential equations of generating functions.
- General Binet's type formulas for various polynomials provided.

## Abstract

The aim of this paper is to construct general forms of ordinary generating functions for special numbers and polynomials involving Fibonacci type numbers and polynomials, Lucas numbers and polynomials, Chebyshev polynomials, Sextet polynomials, Humbert-type numbers and polynomials, chain and anti-chain polynomials, rank polynomials of the lattices, length of any alphabet of words, partitions, and other graph polynomials. By applying the Euler transform and the Lambert series to these generating functions, many new identities and relations are derived. By using differential equations of these generating functions, some new recurrence relations for these polynomials are found. Moreover, general Binet's type formulas for these polynomials are given. Finally, some new classes of polynomials and their corresponding certain family of special numbers are investigated with the help of these generating functions.