Some congruences involving powers of Legendre polynomials

Abstract

The Legendre polynomials $P_n\left(x\right)$ are defined by $P_n\left(x\right)=\sum _{k=0}^n\left(\begin{array}{c}n\\ k\end{array}\right)\left(\begin{array}{c}n+k\\ k\end{array}\right){\left(\frac{x-1}{2}\right)}^k.$ We prove three supercongruences on sums involving $P_k(2x+1{)}^3$ or $P_k(2x+1{)}^4$, two of which were originally conjectured by Z.-W. Sun.

Keywords:

• congruences
• Legendre polynomials
• Clausen's formula
• Zeilberger's algorithm
• Fermat'slittle theorem

MR Subject Classifications:

• 11A07
• 11B65
• 05A10

Acknowledgments

The author thanks Professor Zhi-Wei Sun and the referee for helpful comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was partially supported by the Fundamental Research Funds for the Central Universities and the National Natural Science Foundation of China (grant 11371144).