# J. Number Theory 143(2014), no.3, 293-319. GENERALIZED LEGENDRE POLYNOMIALS AND RELATED SUPERCONGRUENCES

#### Abstract

Abstract. For any positive integer n and variables a and x we define the generalized Legendre polynomial Pn(a, x) by Pn(a, x) = Pn k=0 a k −1−a k ( 1−x 2 ). Let p be an odd prime. In this paper we prove many congruences modulo p related to Pp−1(a, x). For example, we show that Pp−1(a, x) ≡ (−1)〈a〉p Pp−1(a,−x) (mod p), where a is a rational p− adic integer and 〈a〉p is the least nonnegative residue of a modulo p. We also generalize some congruences of Zhi-Wei Sun, and establish congruences for Pp−1 k=0 2k k 3k k Æ 54 and Pp−1 k=0 a k b−a k (mod p).

### Cite this paper

@inproceedings{Sun2014JNT, title={J. Number Theory 143(2014), no.3, 293-319. GENERALIZED LEGENDRE POLYNOMIALS AND RELATED SUPERCONGRUENCES}, author={Zhi-Hong Sun}, year={2014} }