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The Ramanujan Journal - The Gauss summation theorem and an extended $$_3F_2$$ -series of Watson and Whipple type are examined by means of power series expansions. Numerous Ramanujan-like series...  相似文献   

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Some new congruences on harmonic numbers are established. In addition, we obtain a congruence of binomial sums, which is a generalization of that of van Hamme and confirms a conjecture of Swisher.  相似文献   

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In this paper, we consider the generalized Catalan numbers , which we call s-Catalan numbers. For p prime, we find all positive integers n such that pq divides F(pq,n), and also determine all distinct residues of , q?1. As a byproduct we settle a question of Hough and the late Simion on the divisibility of the 4-Catalan numbers by 4. In the second part of the paper we prove that if pq?99999, then is not squarefree for n?τ1(pq) sufficiently large (τ1(pq) computable). Moreover, using the results of the first part, we find n<τ1(pq) (in base p), for which may be squarefree. As consequences, we obtain that is squarefree only for n=1,3,45, and is squarefree only for n=1,4,10.  相似文献   

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In this study, some new properties of Lucas numbers with binomial coefficients have been obtained to write Lucas sequences in a new direct way. In addition, some important consequences of these results related to the Fibonacci numbers have been given.  相似文献   

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We estimate the number of solutions of certain congruences with Catalan numbers and middle binomial coefficients modulo a prime. We use these results to bound double exponential sums with products of two Catalan numbers and two middle binomial coefficients, respectively, which in turn lead us to upper bounds on single exponential sums.  相似文献   

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Hasan Coskun 《Discrete Mathematics》2010,310(17-18):2280-2298
Multiple qt-binomial coefficients and multiple analogues of several celebrated families of related special numbers are constructed in this paper. These higher-dimensional generalizations include the first and the second kind of qt-Stirling numbers, qt-Bell numbers, qt-Bernoulli numbers, qt-Catalan numbers and the qt-Fibonacci numbers. Certain significant applications are also studied including two discrete probability measures on the set of all integer partitions.  相似文献   

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We present various inequalities for the harmonic numbers defined by ${H_n=1+1/2 +\ldots +1/n\,(n\in{\bf N})}$ . One of our results states that we have for all integers n ???2: $$\alpha \, \frac{\log(\log{n}+\gamma)}{n^2} \leq H_n^{1/n} -H_{n+1}^{1/(n+1)} < \beta \, \frac{\log(\log{n}+\gamma)}{n^2}$$ with the best possible constant factors $$\alpha= \frac{6 \sqrt{6}-2 \sqrt[3]{396}}{3 \log(\log{2}+\gamma)}=0.0140\ldots \quad\mbox{and} \quad\beta=1.$$ Here, ?? denotes Euler??s constant.  相似文献   

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In this paper, we prove some congruences conjectured by Z.-W. Sun: For any prime \(p>3\), we determine
$$\begin{aligned} \sum \limits _{k = 0}^{p - 1} {\frac{{{C_k}C_k^{(2)}}}{{{{27}^k}}}} \quad {\text { and }}\quad \sum \limits _{k = 1}^{p - 1} {\frac{{\left( {\begin{array}{l} {2k} \\ {k - 1} \\ \end{array}} \right) \left( { \begin{array}{l} {3k} \\ {k - 1} \\ \end{array} } \right) }}{{{{27}^k}}}} \end{aligned}$$
modulo \(p^2\), where \(C_k=\frac{1}{k+1}\left( {\begin{array}{c}2k\\ k\end{array}}\right) \) is the k-th Catalan number and \(C_k^{(2)}=\frac{1}{2k+1}\left( {\begin{array}{c}3k\\ k\end{array}}\right) \) is the second-order Catalan numbers of the first kind. And we prove that
$$\begin{aligned} \sum _{k=1}^{p-1}\frac{D_k}{k}\equiv -q_p(2)+pq_p(2)^2\pmod {p^2}, \end{aligned}$$
where \(D_n=\sum _{k=0}^{n}\left( {\begin{array}{c}n\\ k\end{array}}\right) \left( {\begin{array}{c}n+k\\ k\end{array}}\right) \) is the n-th Delannoy number and \(q_p(2)=(2^{{p-1}}-1)/p\) is the Fermat quotient.
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Let q>1 and m>0 be relatively prime integers. We find an explicit period νm(q) such that for any integers n>0 and r we have
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We characterize commutative domainsR for which theR-module ofR-valued polynomials is generated by binomial coefficients. This turns out to be a special case of a more general result concerning commutative ringsR of zero characteristics in which fork=1,2,... and allxR the productx(x–1)·­.·(x–k+1) is divisible byk! inR.The work of the second author has been sponsored by the KBN grant 2 1037 91 01  相似文献   

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We prove that if m and \({\nu}\) are integers with \({0 \leq \nu \leq m}\) and x is a real number, then
  1. $$\sum_{k=0 \atop k+m \, \, odd}^{m-1} {m \choose k}{k+m \choose \nu} B_{k+m-\nu}(x) = \frac{1}{2} \sum_{j=0}^m (-1)^{j+m} {m \choose j}{j+m-1 \choose \nu} (j+m) x^{j+m-\nu-1},$$ where B n (x) denotes the Bernoulli polynomial of degree n. An application of (1) leads to new identities for Bernoulli numbers B n . Among others, we obtain
  2. $$\sum_{k=0 \atop k+m \, \, odd}^{m -1} {m \choose k}{k+m \choose \nu} {k+m-\nu \choose j}B_{k+m-\nu-j} =0 \quad{(0 \leq j \leq m-2-\nu)}. $$ This formula extends two results obtained by Kaneko and Chen-Sun, who proved (2) for the special cases j = 1, \({\nu=0}\) and j = 3, \({\nu=0}\) , respectively.
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16.
Renzo Sprugnoli   《Discrete Mathematics》2008,308(22):5070-5077
We extend the concept of a binomial coefficient to all integer values of its parameters. Our approach is purely algebraic, but we show that it is equivalent to the evaluation of binomial coefficients by means of the Γ-function. In particular, we prove that the traditional rule of “negation” is wrong and should be substituted by a slightly more complex rule. We also show that the “cross product” rule remains valid for the extended definition.  相似文献   

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Harmonic numbers and generalized harmonic numbers have been studied since the distant past and involved in a wide range of diverse fields such as analysis of algorithms in computer science, various branches of number theory, elementary particle physics and theoretical physics. Here we aim at presenting further interesting identities about certain finite or infinite series involving harmonic numbers and generalized harmonic numbers by applying an algorithmic method to a known summation formula for the hypergeometric function 5F4(1).  相似文献   

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A general formulae for the asymptotic expansion of not centered binomial coefficients are derived and some useful estimates of the binomial coefficients are presented. The sum of the binomial coefficients is also studied.  相似文献   

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