Supercongruences involving Lucas sequences

Abstract

For $$A,B\\in {\\mathbb {Z}}$$\n , the Lucas sequence $$u_n(A,B)\\ (n=0,1,2,\\ldots )$$\n are defined by $$u_0(A,B)=0$$\n , $$u_1(A,B)=1$$\n , and $$u_{n+1}(A,B)=Au_n(A,B)-Bu_{n-1}(A,B)\\ (n=1,2,3,\\ldots ).$$\n For any odd prime p and positive integer n, we establish the new result $$\\begin{aligned} \\frac{u_{pn}(A,B)-\\left( \\frac{A^2-4B}{p}\\right) u_n(A,B)}{pn}\\in {\\mathbb {Z}}_p, \\end{aligned}$$\n where $$\\left( \\frac{\\cdot }{p}\\right) $$\n is the Legendre symbol and $${\\mathbb {Z}}_p$$\n is the ring of p-adic integers. Let p be an odd prime and let n be a positive integer. For any integer $$m\\not \\equiv 0\\pmod p$$\n , we prove that $$\\begin{aligned} \\frac{1}{pn}\\left( \\sum _{k=0}^{pn-1}\\frac{\\left( {\\begin{array}{c}2k\\\\ k\\end{array}}\\right) }{m^k}-\\left( \\frac{\\Delta }{p}\\right) \\sum _{r=0}^{n-1}\\frac{\\left( {\\begin{array}{c}2r\\\\ r\\end{array}}\\right) }{m^r}\\right) \\in {\\mathbb {Z}}_p \\end{aligned}$$\n and furthermore $$\\begin{aligned} \\frac{1}{n}\\left( \\sum _{k=0}^{pn-1}\\frac{\\left( {\\begin{array}{c}2k\\\\ k\\end{array}}\\right) }{m^k}-\\left( \\frac{\\Delta }{p}\\right) \\sum _{r=0}^{n-1}\\frac{\\left( {\\begin{array}{c}2r\\\\ r\\end{array}}\\right) }{m^r}\\right) \\equiv \\frac{\\left( {\\begin{array}{c}2n-1\\\\ n-1\\end{array}}\\right) }{m^{n-1}}u_{p-\\left( \\frac{\\Delta }{p}\\right) }(m-2,1)\\pmod {p^2}, \\end{aligned}$$\n where $$\\Delta =m(m-4)$$\n . We also pose some conjectures for further research.