有关广义中心三项式系数的3个超同余式

Three Supercongruences Involving Generalized Central Trinomial Coefficients

设b,c为整数,定义广义中心三项式系数T_(n)(b,c)=[x^(n)x^(2)+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/m^(2)p^(3r-3)(p^(r)m-1∑k=0(2k+1)Dk(x)^(2)-P^(2)p^(r-1m-1)∑k=0(2k+1)Dk(x)^(2))=0(mod p^(3)).

For b,c∈Z,the generalized central trinomial coefficients are defined by T_(n)(b,c)=[x^(n)x^(2)+bx+c]^(n)=[π/2]∑k=0(n 2k)(2k n)b^(n-2kck)(n∈N={0,1...}),where[xn]P(x)denotes the coefficient of xn in a polynomial P(x).In particular,those integers Dn(x)=T_(n)(2 x+1,x^(2)+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 in[J.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/m^(2)p^(3r-3)(p^(r)m-1∑k=0(2k+1)Dk(x)^(2)-P^(2)p^(r-1m-1)∑k=0(2k+1)Dk(x)^(2))=0(mod p^(3)).