Preprint Article Version 1 NOT YET PEER-REVIEWED

# Some Properties of the Hermite Polynomials and Their Squares and Generating Functions

Feng Qi 1,2,3,* and Bai-Ni Guo 4
1
Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province 454010, China
2
College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region 028043, China
3
Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City 300387, China
4
School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo City, Henan Province 454010, China
Version 1 : Received: 29 November 2016 / Approved: 29 November 2016 / Online: 29 November 2016 (06:50:11 CET)

How to cite: Qi, F.; Guo, B. Some Properties of the Hermite Polynomials and Their Squares and Generating Functions. Preprints 2016, 2016110145 (doi: 10.20944/preprints201611.0145.v1). Qi, F.; Guo, B. Some Properties of the Hermite Polynomials and Their Squares and Generating Functions. Preprints 2016, 2016110145 (doi: 10.20944/preprints201611.0145.v1).

## Abstract

In the paper, the authors consider the generating functions of the Hermite polynomials and their squares, present explicit formulas for higher order derivatives of the generating functions of the Hermite polynomials and their squares, which can be viewed as ordinary differential equations or derivative polynomials, find differential equations that the generating functions of the Hermite polynomials and their squares satisfy, and derive explicit formulas and recurrence relations for the Hermite polynomials and their squares.

## Subject Areas

Hermite polynomial; square; generating function; higher order derivative; differential equation; derivative polynomial; explicit formula; recurrence relation