共查询到20条相似文献,搜索用时 0 毫秒
1.
Christopher Bingham 《Journal of multivariate analysis》1974,4(2):210-223
Define coefficients (κλ) by Cλ(Ip + Z)/Cλ(Ip) = Σk=0l Σ?∈k (?λ) Cκ(Z)/Cκ(Ip), where the Cλ's are zonal polynomials in p by p matrices. It is shown that C?(Z) etr(Z)/k! = Σl=k∞ Σλ∈l (?λ) Cλ(Z)/l!. This identity is extended to analogous identities involving generalized Laguerre, Hermite, and other polynomials. Explicit expressions are given for all (?λ), ? ∈ k, k ≤ 3. Several identities involving the (?λ)'s are derived. These are used to derive explicit expressions for coefficients of in expansions of P(Z), for all monomials P(Z) in sj = tr Zj of degree k ≤ 5. 相似文献
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C. J. Henrich 《Numerische Mathematik》1975,24(1):81-83
An efficient way to evaluate \(\sum\limits_{j = k}^n {( - 1)^{j - k - 1} \left( {\begin{array}{*{20}c} n \\ j \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {j - 1} \\ {k - 1} \\ \end{array} } \right)} \ln j\) is described. This sum, connected with the logarithmic Weibull distribution, is hard to evaluate directly, because the binomial coefficients become quite large, and then the alternating signs cause severe loss of significant figures. By converting the sum to an integral, we avoid this difficulty. 相似文献
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In the present paper we investigate some problems connected with the positive definiteness of the sequences $j \to e^{j^\alpha } (j \in N_0 )$ and $j \to e^{ - \left| j \right|^\alpha } (j \in Z)$ , whereα≥0. For this we need and prove some results about certain determinants and finite sums that might be of independent interest. 相似文献
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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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In this paper we establish a q-analogue of a congruence of Sun concerning the products of binomial coefficients modulo the square of a prime. 相似文献
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Guo-Shuai Mao 《The Ramanujan Journal》2018,45(2):319-330
In this paper, we prove some congruences conjectured by Z.-W. Sun: For any prime \(p>3\), we determine 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 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.
相似文献
$$\begin{aligned} \sum \limits _{k = 0}^{p - 1} {\frac{{{C_k}C_k^{(2)}}}{{{{27}^k}}}} \quad {\text { and }}\quad \sum \limits _{k = 1}^{p - 1} {\frac{{\left( {\begin{array}{l} {2k} \\ {k - 1} \\ \end{array}} \right) \left( { \begin{array}{l} {3k} \\ {k - 1} \\ \end{array} } \right) }}{{{{27}^k}}}} \end{aligned}$$
$$\begin{aligned} \sum _{k=1}^{p-1}\frac{D_k}{k}\equiv -q_p(2)+pq_p(2)^2\pmod {p^2}, \end{aligned}$$
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Sandro Mattarei 《Journal of Number Theory》2006,117(2):471-481
We prove that if the signed binomial coefficient viewed modulo p is a periodic function of i with period h in the range 0?i?k, then k+1 is a power of p, provided h is not too large compared to k. (In particular, 2h?k suffices). As an application, we prove that if G and H are multiplicative subgroups of a finite field, with H<G, and such that 1-α∈G for all α∈G?H, then G∪{0} is a subfield. 相似文献
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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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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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Christian Krattenthaler 《Monatshefte für Mathematik》1989,107(4):333-339
We give a combinatorial proof that
is a polynomial inq with nonnegative coefficients for nonnegative integersa, b, k, l withab andlk. In particular, fora=b=n andl=k, this implies theq-log-concavity of the Gaussian binomial coefficients
, which was conjectured byButler (Proc. Amer. Math. Soc. 101 (1987), 771–775). 相似文献
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Michel Lassalle 《The Ramanujan Journal》2014,34(1):143-156
Let t be a fixed parameter and x some indeterminate. We give some properties of the generalized binomial coefficients $\genfrac{\langle }{\rangle}{0pt}{}{x}{k}$ inductively defined by $k/x \genfrac{\langle}{\rangle}{0pt}{}{x}{k}= t\genfrac{\langle}{\rangle}{0pt}{}{x-1}{k-1} +(1-t)\genfrac{\langle}{\rangle}{0pt}{}{x-2}{k-2}$ . 相似文献
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It is known that for sufficiently large n and m and any r the binomial coefficient (nm) which is close to the middle coefficient is divisible by pr where p is a ‘large’ prime. We prove the exact divisibility of (nm) by pr for p> c(n). The lower bound is essentially the best possible. We also prove some other results on divisibility of binomial coefficients. 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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Roberto Tauraso 《Journal of Number Theory》2010,130(12):2639-2649