An extension of Sury's identity and related congruences (Q2891977)

This paper proves that for any positive integer \(n\), letting \(1\leq j \leq n+1\) be odd, being \(H_n=\sum_{k=1}^n 1/k\) the \(n\)th harmonic number (with \(H_0=0\)) and denoting by \(A\), \(B\), \(C\), \(D\), \(E\) the following expressions NEWLINE\[NEWLINE A = \frac{2^n}{n+1} \cdot \sum_{k=0}^{n} \frac {1}{\binom n k}, NEWLINE\]NEWLINE NEWLINE\[NEWLINE B = \sum_{k=0}^{n} \frac{2^k}{k+1}, NEWLINE\]NEWLINE NEWLINE\[NEWLINE C = \sum_{j=1}^{n+1} \frac{1}{j} \binom{n+1}{j}, NEWLINE\]NEWLINE NEWLINE\[NEWLINE D = \frac{1}{2} \sum_{i=1}^{n+1} \frac{1}{i} \binom{n+1}{i} + \frac{1}{2} H_{n+1}, NEWLINE\]NEWLINE NEWLINE\[NEWLINE E = 2^n H_{n+1} - \sum_{k=1}^{n} 2^{k-1} H_k , NEWLINE\]NEWLINE we have \(A=B=C=D=E\).NEWLINENEWLINEIt extends a combinatorial identity proved by \textit{B. Sury} [Eur. J. Comb. 14, No. 4, 351--353 (1993; Zbl 0783.05002)] and related to an old work by \textit{T. B. Staver} [Norsk Mat. Tidsskr. 29, 97--103 (1947; Zbl 0030.28901)]. As a corollary, it is established another identity already supplied by \textit{B. Sury, T. Wang} and \textit{F.-Z. Zhao} [J. Integer Seq. 7, No. 2, Art. 04.2.8, 12 p. (2004; Zbl 1069.11008)].NEWLINENEWLINENEWLINEAs an application, the author obtains two new congruences for Fermat quotients modulo \(p^3\) related to some results by \textit{J. W. L. Glaisher} [Q. J. Math., Oxf. 32, 271--288 (1900; JFM 31.0186.03)], by \textit{A. Granville} [Integers 4, Paper A22, 3 p. (2004; Zbl 1083.11005)], by \textit{Z.-W. Sun} [J. Number Theory 128, No. 2, 280--312 (2008; Zbl 1154.11010)] and by \textit{Z.-W. Sun} [Sci. China, Math. 53, No. 9, 2473--2488 (2010; Zbl 1221.11054)].NEWLINENEWLINENEWLINEThis paper also proves an extension of the generalization of a well-known congruence by \textit{L. Carlitz} [Can. J. Math. 5, 306--316 (1953; Zbl 0052.03802)] due to \textit{Hao Pan} [Int. J. Mod. Math. 4, No. 1, 87--93 (2009; Zbl 1247.11025)] and related to some results by \textit{F. Morley} [Ann. Math. 9, 168--170 (1895; JFM 26.0208.02)] and by \textit{T.-X. Cai} and \textit{A. Granville} [Acta Math. Sin., Engl. Ser. 18, No.2, 277--288 (2002; Zbl 1026.11005)].NEWLINENEWLINENEWLINEIn the proof the author employs the method of induction, Fermat's Little Theorem, some identities illustrated by [\textit{H. W. Gould}, Combinatorial identities. Morgantown, W. Va.: Henry E. Gould (1972; Zbl 0241.05011)], some harmonic number identities provided by \textit{Z.-W. Sun} [Proc. Am. Math. Soc. 140, No. 2, 415--428 (2012; Zbl 1271.11021)] and by \textit{R. Tauraso} [Sémin. Lothar. Comb. 63, B63g, 8 p. (2010; Zbl 1273.11007)] and some congruences concerning the Bernoulli numbers \(B_0, B_1, B_2,\dots\) given by \textit{Z. Sun} [Discrete Appl. Math. 105, No. 1-3, 193--223 (2000; Zbl 0990.11008)].NEWLINENEWLINEFor the interested reader the author suggests other remarkable studies involving reciprocals of binomial coefficients such as the papers by \textit{F.-Z. Zhao} and \textit{T. Wang} [Integers 5, No. 1, Paper A22, 5 p. (2005; Zbl 1085.11008)], by \textit{J.-H. Yang} and \textit{F.-Z. Zhao} [J. Integer Seq. 9, No. 4, Article 06.4.2, 11 p. (2006; Zbl 1108.11022) and J. Integer Seq. 10, No. 8, Article 07.8.7, 11 p. (2007; Zbl 1146.11010)].NEWLINENEWLINEAlthough not cited in the text, the masterpiece by [\textit{G. H. Hardy} and \textit{E. M. Wright}, An introduction to the theory of numbers. 5th ed. Oxford etc.: Oxford at the Clarendon Press (1979; Zbl 0423.10001)] has however been inserted in the reference list, probably as a first introduction to Number Theory.