有关广义中心三项式系数的3个超同余式

设b,c为整数,定义广义中心三项式系数T_n(b,c)=xⁿx²+bx+c]ⁿ=π/2]∑k=0(n 2k)(2k n)b^n-2kck(n∈N={0,1...}),这里xn]P(x)表示多项式P(x)中xn项的系数.特别地,中心Delannoy多项式Dn(x)=T_n(2x+1,x2+x)(n∈N),中心三项式系数T_n=T_n(1,1)(n∈N).本文研究了孙智伟在南京大学学报:数学半年刊,2019,36(1):1-99]中提出的猜想,即完全证明了两个关于Dn(x)和的超同余式和一个关于中心三项式系数的超同余式的特殊情形.例如,设p为素数,r,m为正整数满足p■m条件.则对于任何p-adic整数x,有1/m²p^3r-3(p^rm-1∑k=0(2k+1)Dk(x)²-P²p^r-1m-1∑k=0(2k+1)Dk(x)²)=0(mod p³).

Three Supercongruences Involving Generalized Central Trinomial Coefficients

For b,c∈Z,the generalized central trinomial coefficients are defined by T_n(b,c)=xⁿx²+bx+c]ⁿ=π/2]∑k=0(n 2k)(2k n)b^n-2kck(n∈N={0,1...}),wherexn]P(x)denotes the coefficient of xn in a polynomial P(x).In particular,those integers Dn(x)=T_n(2 x+1,x²+x)(n∈N)and T_n=T_n(1,1)(n∈N)are called central Delannoy polynomials and central trinomial coefficients,respectively.In this paper we prove two supercongruences on sums involving Dn(x)and confirm a special case of a supercongruence involving T_n,which were conjectured by Sun inJ.Nanjing Univ.Math.Biquarterly,2019,36(1):1-99].For example,suppose that p is a prime and r,m∈Z+with p■m.Then for any p-adic integer x,we show that1/m²p^3r-3(p^rm-1∑k=0(2k+1)Dk(x)²-P²p^r-1m-1∑k=0(2k+1)Dk(x)²)=0(mod p³).