On sums of binomial coefficients modulo p^2

Abstract

Let p be an odd prime and let a be a positive integer. In this paper we investigate the sum ${\sum }_{k=0}^{p^a-1}\left(\genfrac{}{}{0ex}{}{h{p}^a-1}{k}\right)\left(\genfrac{}{}{0ex}{}{2k}{k}\right)/m^k$ mod p^2, where h,m are p-adic integers with m\not=0 (mod p). For example, we show that if h\not=0 (mod p) and p^a>3 then $$su{m}_{k=0}^{p^a-1}\left(\genfrac{}{}{0ex}{}{h{p}^a-1}{k}\right)\left(\genfrac{}{}{0ex}{}{2k}{k}\right)(-h/2{)}^k=(\frac{1-2h}{p^a})(1+h((4h-2{)}^{p-1}/h^{p-1}-1))(mod{p}^2),$$ where (-) denotes the Jacobi symbol. Here is another remarkable congruence: If p>3 then $$\sum _{k=0}^{p^a-1}\left(\genfrac{}{}{0ex}{}{p^a-1}{k}\right)\left(\genfrac{}{}{0ex}{}{2k}{k}\right)(-1{)}^k=3^{p-1}(\frac{p^a}{3})(mod{p}^2).$$

13 pages, polished version

References in corpus (2)

• A generalization of Carlitz's congruence
• p-adic valuations of some sums of multinomial coefficients

Cited by in corpus (8)

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• Congruences involving binomial coefficients and Lucas sequences
• Two congruences involving harmonic numbers with applications
• Curious congruences for Fibonacci numbers
• Some congruences involving binomial coefficients