- Publication venue
- 'Springer Science and Business Media LLC'
- Publication date
- 01/01/2010
- Field of study
Full text linkLet p be an odd prime and let a,m be integers with a>0 and mî€ â‰¡0(modp). In this paper we determine
∑k=0pa−1​(k+d2k​)/mk mod p2 for d=0,1; for example,
k=0∑pa−1​mk(k2k​)​≡(pam2−4m​)+(pa−1m2−4m​)up−(pm2−4m​)​(modp2),
where (−) is the Jacobi symbol, and {un​}n⩾0​ is the Lucas
sequence given by u0​=0, u1​=1 and un+1​=(m−2)un​−un−1​ for
n=1,2,3,…. As an application, we determine ∑0<k<pa,k≡r(modp−1)​Ck​ modulo p2 for any integer r, where Ck​ denotes the
Catalan number (k2k​)/(k+1). We also pose some related conjectures.Comment: 24 pages. Correct few typo - Publication venue
- 'Mathematical Association of America'
- Publication date
- Field of study
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