11,101 research outputs found
Congruences concerning Legendre polynomials
Let be an odd prime. In the paper, by using the properties of Legendre
polynomials we prove some congruences for
. In particular, we
confirm several conjectures of Z.W. Sun. We also pose 13 conjectures on
supercongruences.Comment: 16 page
Generalized Legendre polynomials and related congruences modulo
For any positive integer and variables and we define the
generalized Legendre polynomial P_n(a,x)=\sum_{k=0}^n\b
ak\b{-1-a}k(\frac{1-x}2)^k. Let be an odd prime. In the paper we prove
many congruences modulo related to . For example, we show
that P_{p-1}(a,x)\e (-1)^{_p}P_{p-1}(a,-x)\mod {p^2}, where is the
least nonnegative residue of modulo . We also generalize some
congruences of Zhi-Wei Sun, and determine
and , where is the greatest integer function.
Finally we pose some supercongruences modulo concerning binary quadratic
forms.Comment: 37 page
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