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1.
Victor Kowalenko 《Acta Appl Math》2009,106(3):369-420
Recently, a novel method based on the coding of partitions was used to determine a power series expansion for the reciprocal
of the logarithmic function, viz. z/ln (1+z). Here we explain how this method can be adapted to obtain power series expansions for other intractable functions. First,
the method is adapted to evaluate the Bernoulli numbers and polynomials. As a result, new integral representations and properties
are determined for the former. Then via another adaptation of the method we derive a power series expansion for the function
z
s
/ln
s
(1+z), whose polynomial coefficients A
k
(s) are referred to as the generalized reciprocal logarithm numbers because they reduce to the reciprocal logarithm numbers
when s=1. In addition to presenting a general formula for their evaluation, this paper presents various properties of the generalized
reciprocal logarithm numbers including general formulas for specific values of s, a recursion relation and a finite sum identity. Other representations in terms of special polynomials are also derived for
the A
k
(s), which yield general formulas for the highest order coefficients. The paper concludes by deriving new results involving
infinite series of the A
k
(s) for the Riemann zeta and gamma functions and other mathematical quantities. 相似文献
2.
A. I. Vinogradov 《Journal of Mathematical Sciences》2006,137(2):4617-4633
The spectral decomposition for the square of the classical Riemann zeta function ζ2(s) is generalized to the case of the product of two such functions ζ(s1) · ζ(s2) of different arguments. Bibliography: 6 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 322, 2005, pp. 17–44. 相似文献
3.
Meixner polynomials m
n
(x;β,c) form a postive-definite orthogonal system on the positive real line x > 0 with respect to a distribution step function whose jumps are Unlike classical orthogonal polynomials, they do not satisfy a second-order linear differential equation. In this paper,
we derive two infinite asymptotic expansions for m
n
(nα;β,c) as . One holds uniformly for , and the other holds uniformly for , where a and b are two small positive quantities. Both expansions involve the parabolic cylinder function and its derivative. Our results
include all five asymptotic formulas recently given by W. M. Y. Goh as special cases.
April 16, 1996. Date revised: October 30, 1996. 相似文献
4.
Akio Fujii 《Proceedings of the Steklov Institute of Mathematics》2012,276(1):51-76
Let ζ′(s) be the derivative of the Riemann zeta function ζ(s). A study on the value distribution of ζ′(s) at the non-trivial zeros ρ of ζ(s) is presented. In particular, for a fixed positive number X, an asymptotic formula and a non-trivial upper bound for the sum Σ0<Im ρ≤T
ζ′(ρ)X
ρ
as T → ∞ are given. We clarify the dependence on the arithmetic nature of X. 相似文献
5.
Donald St. P. Richards 《Annals of the Institute of Statistical Mathematics》1982,34(1):119-121
Summary LetC
κ(S) be the zonal polynomial of the symmetricm×m matrixS=(sij), corresponding to the partition κ of the non-negative integerk. If ∂/∂S is them×m matrix of differential operators with (i, j)th entry ((1+δij)∂/∂sij)/2, δ being Kronecker's delta, we show that Ck(∂/∂S)Cλ(S)=k!δλkCk(I), where λ is a partition ofk. This is used to obtain new orthogonality relations for the zonal polynomials, and to derive expressions for the coefficients
in the zonal polynomial expansion of homogenous symmetric polynomials. 相似文献
6.
Linas Vepštas 《Numerical Algorithms》2008,47(3):211-252
This paper sketches a technique for improving the rate of convergence of a general oscillatory sequence, and then applies
this series acceleration algorithm to the polylogarithm and the Hurwitz zeta function. As such, it may be taken as an extension
of the techniques given by Borwein’s “An efficient algorithm for computing the Riemann zeta function” by Borwein for computing
the Riemann zeta function, to more general series. The algorithm provides a rapid means of evaluating Li
s
(z) for general values of complex s and a kidney-shaped region of complex z values given by ∣z
2/(z–1)∣<4. By using the duplication formula and the inversion formula, the range of convergence for the polylogarithm may be
extended to the entire complex z-plane, and so the algorithms described here allow for the evaluation of the polylogarithm for all complex s and z values. Alternatively, the Hurwitz zeta can be very rapidly evaluated by means of an Euler–Maclaurin series. The polylogarithm
and the Hurwitz zeta are related, in that two evaluations of the one can be used to obtain a value of the other; thus, either
algorithm can be used to evaluate either function. The Euler–Maclaurin series is a clear performance winner for the Hurwitz
zeta, while the Borwein algorithm is superior for evaluating the polylogarithm in the kidney-shaped region. Both algorithms
are superior to the simple Taylor’s series or direct summation. The primary, concrete result of this paper is an algorithm
allows the exploration of the Hurwitz zeta in the critical strip, where fast algorithms are otherwise unavailable. A discussion
of the monodromy group of the polylogarithm is included.
相似文献
7.
An infinite asymptotic expansion is derived for the Meixner—Pollaczek polynomials M
n
(nα;δ, η) as n→∞ , which holds uniformly for -M≤α≤ M , where M can be any positive number. This expansion involves the parabolic cylinder function and its derivative. If α
n, s
denotes the s th zero of M
n
(nα;δ, η) , counted from the right, and if α˜
n,s
denotes its s th zero counted from the left, then for each fixed s , three-term asymptotic approximations are obtained for both α
n,s
and α˜
n,s
as n→∞ .
December 28, 1998. Date revised: June 4, 1999. Date accepted: September 6, 1999. 相似文献
8.
Stanisław Lewanowicz Eduardo Godoy Iván Area André Ronveaux Alejandro Zarzo 《Numerical Algorithms》2000,23(1):31-50
We propose an algorithm to construct recurrence relations for the coefficients of the Fourier series expansions with respect
to the q-classical orthogonal polynomials pk(x;q). Examples dealing with inversion problems, connection between any two sequences of q-classical polynomials, linearization
of ϑm(x) pn(x;q), where ϑm(x) is xmor (x;q)m, and the expansion of the Hahn-Exton q-Bessel function in the little q-Jacobi polynomials are discussed in detail.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
9.
Michael O. Rubinstein 《The Ramanujan Journal》2013,32(3):421-464
We derive several identities for the Hurwitz and Riemann zeta functions, the Gamma function, and Dirichlet L-functions. They involve a sequence of polynomials α k (s) whose study was initiated in Rubinstein (Ramanujan J. 27(1): 29–42, 2012). The expansions given here are practical and can be used for the high precision evaluation of these functions, and for deriving formulas for special values. We also present a summation formula and use it to generalize a formula of Hasse. 相似文献
10.
Expansions in terms of Bessel functions are considered of the Kummer function 1
F
1(a; c, z) (or confluent hypergeometric function) as given by Tricomi and Buchholz. The coefficients of these expansions are polynomials
in the parameters of the Kummer function and the asymptotic behavior of these polynomials for large degree is given. Tables
are given to show the rate of approximation of the asymptotic estimates. The numerical performance of the expansions is discussed
together with the numerical stability of recurrence relations to compute the polynomials. The asymptotic character of the
expansions is explained for large values of the parameter a of the Kummer function. 相似文献
11.
We fix a prime p and let f(X) vary over all monic integer polynomials of fixed degree n. Given any possible shape of a tamely ramified splitting of p in an extension of degree n, we prove that there exists a rational function φ(X)∈ℚ(X) such that the density of the monic integer polynomials f(X) for which the splitting of p has the given shape in ℚ[X]/f(X) is φ(p) (here reducible polynomials can be neglected). As a corollary, we prove that, for p≥n, the density of irreducible monic polynomials of degree n in ℤ
p
[X] is the value at p of a rational function φ
n
(X)∈ℚ(X). All rational functions involved are effectively computable.
Received: 15 September 1998 / Revised version: 21 October 1999 相似文献
12.
Junesang Choi P.J. Anderson H.M. Srivastava 《Applied mathematics and computation》2009,215(3):1185-1208
In this paper, we systematically recover the identities for the q-eta numbers ηk and the q-eta polynomials ηk(x), presented by Carlitz [L. Carlitz, q-Bernoulli numbers and polynomials, Duke Math. J. 15 (1948) 987–1000], which we define here via generating series rather than via the difference equations of Carlitz. Following a method developed by Kaneko et al. [M. Kaneko, N. Kurokawa, M. Wakayama, A variation of Euler’s approach to the Riemann zeta function, Kyushu J. Math. 57 (2003) 175–192] for a canonical q-extension of the Riemann zeta function, we investigate a similarly constructed q-extension of the Hurwitz zeta function. The details of this investigation disclose some interesting connections among q-eta polynomials, Carlitz’s q-Bernoulli polynomials -polynomials, and the q-Bernoulli polynomials that emerge from the q-extension of the Hurwitz zeta function discussed here. 相似文献
13.
To investigate the quality of heavy-traffic approximations for queues with many servers, we consider the steady-state number
of waiting customers in an M/D/s queue as s→∞. In the Halfin-Whitt regime, it is well known that this random variable converges to the supremum of a Gaussian random
walk. This paper develops methods that yield more accurate results in terms of series expansions and inequalities for the
probability of an empty queue, and the mean and variance of the queue length distribution. This quantifies the relationship
between the limiting system and the queue with a small or moderate number of servers. The main idea is to view the M/D/s queue through the prism of the Gaussian random walk: as for the standard Gaussian random walk, we provide scalable series
expansions involving terms that include the Riemann zeta function.
相似文献
14.
N. N. Pustovoitov 《Mathematical Notes》1999,65(1):89-98
In the paper order-exact upper bounds for the best approximations of classesH
q
Emphasis>/ω by trigonometric polynomials are obtained. The spectrum of the approximating polynomials lies in sets generated by the level
surfaces of the function ω(t). These sets are a generalization of hyperbolic crosses to the case of an arbitrary function ω(t).
Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 107–117, January, 1999. 相似文献
15.
16.
In the study of differential equations on [ − 1,1] subject to linear homogeneous boundary conditions of finite order, it is
often expedient to represent the solution in a Galerkin expansion, that is, as a sum of basis functions, each of which satisfies
the given boundary conditions. In order that the functions be maximally distinct, one can use the Gram-Schmidt method to generate
a set orthogonal with respect to a particular weight function. Here we consider all such sets associated with the Jacobi weight
function, w(x) = (1 − x)
α
(1 + x)
β
. However, this procedure is not only cumbersome for sets of large degree, but does not provide any intrinsic means to characterize
the functions that result. We show here that each basis function can be written as the sum of a small number of Jacobi polynomials,
whose coefficients are found by imposing the boundary conditions and orthogonality to the first few basis functions only.
That orthogonality of the entire set follows—a property we term “auto-orthogonality”—is remarkable. Additionally, these basis
functions are shown to behave asymptotically like individual Jacobi polynomials and share many of the latter’s useful properties.
Of particular note is that these basis sets retain the exponential convergence characteristic of Jacobi expansions for expansion
of an arbitrary function satisfying the boundary conditions imposed. Further, the associated error is asymptotically minimized
in an L
p(α) norm given the appropriate choice of α = β. The rich algebraic structure underlying these properties remains partially obscured by the rather difficult form of the
non-standard weighted integrals of Jacobi polynomials upon which our analysis rests. Nevertheless, we are able to prove most
of these results in specific cases and certain of the results in the general case. However a proof that such expansions can
satisfy linear boundary conditions of arbitrary order and form appears extremely difficult. 相似文献
17.
Extending the problem of determining Ramsey numbers Erdős and Rogers introduced the following function. For given integers
2 ≤ s < t let f
s,t
(n) = min{max{|S|: S ⊆ V (H) and H[S] contains no K
s
}}, where the minimum is taken over all K
t
-free graphs H of order n. This function attracted a considerable amount of attention but despite that, the gap between the lower and upper bounds
is still fairly wide. For example, when t=s+1, the best bounds have been of the form Ω(n
1/2+o(1)) ≤ f
s,s+1(n) ≤ O(n
1−ɛ(s)), where ɛ(s) tends to zero as s tends to infinity. In this paper we improve the upper bound by showing that f
s,s+1(n) ≤ O(n
2/3). Moreover, we show that for every ɛ > 0 and sufficiently large integers 1 ≪ k ≪ s, Ω(n
1/2−ɛ
) ≤ f
s,s+k
(n) ≤ O(n
1/2+ɛ
. In addition, we also discuss some connections between the function f
s,t
and vertex Folkman numbers. 相似文献
18.
J. Dippon 《Mathematical Methods of Statistics》2008,17(2):138-145
Assume that the function values f(x) of an unknown regression function f: ℝ → ℝ can be observed with some random error V. To estimate the zero ϑ of f, Robbins and Monro suggested to run the recursion X
n+1 = X
n
− a/n
Y
n
with Y
n
= f(X
n
) − V
n
. Under regularity assumptions, the normalized Robbins-Monro process, given by (X
n+1 − ϑ)/√Var(X
n+1, is asymptotically standard normal. In this paper Edgeworth expansions are presented which provide approximations of the
distribution function up to an error of order o(1/√n) or even o(1/n). As corollaries asymptotic confidence intervals for the unknown parameter ϑ are obtained with coverage probability errors of order O(1/n). Further results concern Cornish-Fisher expansions of the quantile function, an Edgeworth correction of the distribution
function and a stochastic expansion in terms of a bivariate polynomial in 1/√n and a standard normal random variable. The proofs of this paper heavily rely on recently published results on Edgeworth expansions
for approximations of the Robbins-Monro process.
相似文献
19.
A multiplication theorem for the Lerch zeta function ?(s,a,ξ) is obtained, from which, when evaluating at s=−n for integers n?0, explicit representations for the Bernoulli and Euler polynomials are derived in terms of two arrays of polynomials related to the classical Stirling and Eulerian numbers. As consequences, explicit formulas for some special values of the Bernoulli and Euler polynomials are given. 相似文献
20.
Winfried Kohnen 《Proceedings Mathematical Sciences》1989,99(3):231-233
The conjecture is made that the rational structures on spaces of modular forms coming from the rationality of Fourier coefficients
and the rationality of periods are not compatible. A consequence would be that ζ(2k-1)/π
2k-1 (ζ(s)=Riemann zeta function;k∈ℕ,k≥2) is irrational or even transcendental. 相似文献