Congruences concerning generalized central trinomial coefficients

For any n ∈ N = { 0 , 1 2 …<!-- … stretchy="false">} n\in \mathbb {N}=\{0,1,2,\ldots \} and alttext="b c Z"> b c mathvariant="double-struck">Z encoding="application/x-tex">b,c\in {Z} , the generalized central trinomial coefficient alttext="upper T Subscript n Baseline left-parenthesis b right-parenthesis"> T stretchy="false">( stretchy="false">) encoding="application/x-tex">T_n(b,c) denotes of alttext="x Superscript n"> x encoding="application/x-tex">x^n in expansion alttext="left-parenthesis x squared plus right-parenthesis + encoding="application/x-tex">(x^2+bx+c)^n . Let alttext="p"> p encoding="application/x-tex">p be an odd prime. In this paper, we determine summations alttext="sigma-summation Underscript k Overscript p minus Endscripts slash m k"> ∑<!-- ∑ <mml:mi>k −<!-- − </mml:munderover> / m encoding="application/x-tex">\sum _{k=0}^{p-1}T_k(b,c)^2/m^k modulo alttext="p squared"> encoding="application/x-tex">p^2 for integers alttext="m"> encoding="application/x-tex">m with certain restrictions. As applications, confirm some conjectural congruences Sun [Sci. China Math. 57 (2014), pp. 1375–1400].