共查询到20条相似文献,搜索用时 32 毫秒
1.
We prove that, for all integers \(n\ge 1\), and with the best possible constants
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$$\begin{aligned} \Big (\sqrt{2\pi n}\Big )^{\frac{1}{n(n+1)}}\left( 1-\frac{1}{n+a}\right) <\frac{\root n \of {n!}}{\root n+1 \of {(n+1)!}}\le \Big (\sqrt{2\pi n}\Big )^{\frac{1}{n(n+1)}}\left( 1-\frac{1}{n+b}\right) \end{aligned}$$
$$\begin{aligned} \big (\sqrt{2\pi n}\big )^{1/n}\left( 1-\frac{1}{2n+\alpha }\right) <\left( 1+\frac{1}{n}\right) ^{n}\frac{\root n \of {n!}}{n}\le \big (\sqrt{2\pi n}\big )^{1/n}\left( 1-\frac{1}{2n+\beta }\right) , \end{aligned}$$
$$\begin{aligned}&a=\frac{1}{2},\quad b=\frac{1}{2^{3/4}\pi ^{1/4}-1}=0.807\ldots ,\quad \alpha =\frac{13}{6} \\&\text {and}\quad \beta =\frac{2\sqrt{2}-\sqrt{\pi }}{\sqrt{\pi }-\sqrt{2}}=2.947\ldots . \end{aligned}$$
2.
We consider the question of evaluating the normalizing multiplier $$\gamma _{n,k} = \frac{1}{\pi }\int_{ - \pi }^\pi {\left( {\frac{{sin\tfrac{{nt}}{2}}}{{sin\tfrac{t}{2}}}} \right)^{2k} dt} $$ for the generalized Jackson kernel J n,k (t). We obtain the explicit formula $$\gamma _{n,k} = 2\sum\limits_{p = 0}^{\left[ {k - \tfrac{k}{n}} \right]} {( - 1)\left( {\begin{array}{*{20}c} {2k} \\ p \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {k(n + 1) - np - 1} \\ {k(n - 1) - np} \\ \end{array} } \right)} $$ and the representation $$\gamma _{n,k} = \sqrt {\frac{{24}}{\pi }} \cdot \frac{{(n - 1)^{2k - 1} }}{{\sqrt {2k - 1} }}\left[ {1\frac{1}{8} \cdot \frac{1}{{2k - 1}} + \omega (n,k)} \right],$$ , where $$\left| {\omega (n,k)} \right| < \frac{4}{{(2k - 1)\sqrt {ln(2k - 1)} }} + \sqrt {12\pi } \cdot \frac{{k^{\tfrac{3}{2}} }}{{n - 1}}\left( {1 + \frac{1}{{n - 1}}} \right)^{2k - 2} .$$ . 相似文献
3.
Horst Alzer 《Mediterranean Journal of Mathematics》2008,5(4):395-413
We present several sharp inequalities for the volume of the unit ball in ,
. One of our theorems states that the double-inequality
holds for all n ≥ 2 with the best possible constants
This refines and complements a result of Klain and Rota.
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4.
Filip Saidak 《Archiv der Mathematik》2005,85(4):345-361
Assuming a quasi Generalized Riemann Hypothesis (quasi-GRH for short) for Dedekind zeta functions over Kummer fields of the
type
we prove the following prime analogue of a conjecture of Erd?s & Pomerance (1985) concerning the exponent function fa(p) (defined to be the minimal exponent e for which ae ≡ 1 modulo p):
where
The main result is obtained by computing all the higher moments corresponding to ω(fa(p)), and by comparing them, via the Fréchet-Shohat theorem, with estimates due to Halberstam concerning the moments of ω(p − 1).
Received: 25 October 2004; revised: 12 February 2005 相似文献
((‡)) |
5.
Kenji Nishihara 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2006,57(4):604-614
We consider the Cauchy problem for the nonlinear dissipative evolution system with ellipticity on one dimensional space
with
S. Q. Tang and H. Zhao [4] have considered the problem and obtained the optimal decay property for suitably small data. In
this paper we derive the asymptotic profile using the Gauss kernel G(t, x), which shows the precise behavior of solution as time tends to infinity. In fact, we will show that the asymptotic formula
holds, where D0, β0 are determined by the data. It is the key point to reformulate the system to the nonlinear parabolic one by suitable changing
variables.
(Received: January 8, 2005) 相似文献
6.
V. M. Babich 《Journal of Mathematical Sciences》2006,132(1):2-10
The paper is devoted to a detailed consideration of an ansatz known from the seventies:
where
Here the Dp are parabolic-cylinder functions. Analytic expressions in the first approximation for the wave field in the penumbra of the
wave reflected by an impedance or transparent cone are obtained. Bibliography: 11 titles.
Dedicated to P. V. Krauklis on the occasion of his seventieth birthday
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 308, 2004, pp. 9–22. 相似文献
7.
8.
Anastasios D. Simalarides 《Periodica Mathematica Hungarica》2000,40(1):71-75
Let be a primitive character mod k, k > 2. In [1], the following elementary estimate
was given, where
by definition. In the present note we sharpen this estimate by a factor 3/4 in the case of an even primitive character , by improving upon the proof given in [1] in a way which does not alter the elementary character of the method. 相似文献
9.
A new Stirling series as continued fraction 总被引:1,自引:0,他引:1
Cristinel Mortici 《Numerical Algorithms》2011,56(1):17-26
We introduce the following new Stirling series as a continued fraction, which is faster than the classical Stirling series.
相似文献
$ n!\sim \sqrt{2\pi n}\left( \frac{n}{e}\right) ^{n}\exp \frac{1}{12n+\frac{ \frac{2}{5}}{n+\frac{\frac{53}{210}}{n+\frac{\frac{195}{371}}{n+\frac{\frac{ 22,\!999}{22,\!737}}{n+\ddots}}}}}, $
10.
Doron S. Lubinsky 《Constructive Approximation》2016,43(2):311-336
Let \(\mu \) and \(\nu \) be measures supported on \(\left( -1,1\right) \) with corresponding orthonormal polynomials \(\left\{ p_{n}^{\mu }\right\} \) and \( \left\{ p_{n}^{\nu }\right\} \), respectively. Define the mixed kernel We establish scaling limits such as where \(S\left( t\right) =\frac{\sin t}{t}\) is the sinc kernel, and \(B\left( \xi \right) \) depends on \({\mu },\nu \) and \(\xi \). This reduces to the classical universality limit in the bulk when \(\mu =\nu \). We deduce applications to the zero distribution of \(K_{n}^{{\mu },\nu }\), and asymptotics for its derivatives.
相似文献
$$\begin{aligned} K_{n}^{{\mu },\nu }\left( x,y\right) =\sum _{j=0}^{n-1}p_{j}^{\mu }\left( x\right) p_{j}^{\nu }\left( y\right) . \end{aligned}$$
$$\begin{aligned}&\lim _{n\rightarrow \infty }\frac{\pi \sqrt{1-\xi ^{2}}\sqrt{\mu ^{\prime }\left( \xi \right) \nu ^{\prime }\left( \xi \right) }}{n}K_{n}^{\mu ,\nu }\left( \xi +\frac{a\pi \sqrt{1-\xi ^{2}}}{n},\xi +\frac{b\pi \sqrt{1-\xi ^{2}}}{n}\right) \\&\quad =S\left( \frac{\pi \left( a-b\right) }{2}\right) \cos \left( \frac{\pi \left( a-b\right) }{2}+B\left( \xi \right) \right) , \end{aligned}$$
11.
Horst Alzer 《Advances in Computational Mathematics》2010,33(3):349-379
We present various inequalities for the error function. One of our theorems states: Let α?≥?1. For all x,y?>?0 we have $$ \delta_{\alpha} < \frac{ \mbox{erf} \left( x+ \mbox{erf}(y)^{\alpha}\right) +\mbox{erf}\left( y+ \mbox{erf}(x)^{\alpha}\right) } {\mbox{erf}\left( \mbox{erf}(x)+\mbox{erf}(y)\right) } < \Delta_{\alpha} $$ with the best possible bounds $$ \delta_{\alpha}= \left\{ \begin{array}{ll} 1+\sqrt{\pi}/2, & \ \ \textrm{{if} $\alpha=1$,}\\ \sqrt{\pi}/2, & \ \ \textrm{{if} $\alpha>1$,}\\ \end{array}\right. \quad{\mbox{and} \,\,\,\,\, \Delta_{\alpha}=1+\frac{1}{\mbox{erf}(1)}.} $$ 相似文献
12.
For a bounded function defined on
, let
be the set of singular values of the (n + 1) x (n + 1) matrix whose (j, k)-entries
are equal to
These matrices can be thought of as variable-coefficient Toeplitz matrices or
generalized Toeplitz matrices. Matrices of the above form can be also thought
of as the discrete analogue of pseudodifferential operators. Under a certain
smoothness assumption on the function , we prove that
where the constant c1 and a part of c2 are shown to have explicit integral
representations. The other part of c2 turns out to have a resemblance to the
Toeplitz case. This asymptotic formula can be viewed as a generalization of
the classical theory on singular values of Toeplitz matrices. 相似文献
13.
A. A. Abilov 《Mathematical Notes》1992,52(1):631-635
Let
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14.
Xiaojing Yang 《Archiv der Mathematik》2005,85(5):460-469
In this paper, the existence of unbounded solutions for the following nonlinear asymmetric oscillator
15.
We consider the following Liouville equation in
16.
J.-P. Allouche 《The Ramanujan Journal》2007,14(1):39-42
We answer a question of Berndt and Bowman, asking whether it is possible to deduce the value of the Ramanujan integral I from the value of the Ramanujan integral J, where
17.
Aleksandar Ivić 《Central European Journal of Mathematics》2005,3(2):203-214
Let Δ(x) denote the error term in the Dirichlet divisor problem, and E(T) the error term in the asymptotic formula for the mean square of
. If E
*(t)=E(t)-2πΔ*(t/2π) with
, then we obtain
18.
We prove the following statement.
Let , and let . Suppose that, for all and , the sequence satisfies the relation
19.
Gabor Hetyei 《Discrete and Computational Geometry》2006,35(3):437-455
We introduce a new encoding of the face numbers of a simplicial complex, its Stirling polynomial, that has a simple expression
obtained by multiplying each face number with an appropriate generalized binomial coefficient. We prove that the face numbers
of the barycentric subdivision of the free join of two CW-complexes may be found by multiplying the Stirling polynomials of
the barycentric subdivisions of the original complexes. We show that the Stirling polynomial of the order complex of any simplicial
poset and of certain graded planar posets has non-negative coefficients. By calculating the Stirling polynomial of the order
complex of the r-cubical lattice of rank n + 1 in two ways, we provide a combinatorial proof for the following identity of
Bernoulli polynomials:
20.
In studying local harmonic analysis on the sphere Sn, R.S. Strichartz introduced certain zonal functions ϕ2(d(x, y)) which satisfy the equation
, where Δz is the Laplace operator and δ−y the Dirac measure. The explicit expression of the constant a (λ) is given by R.S. Strichartz in the case that n is odd. Appyling
the Apéry identity, we show in this paper that
for n even, where wn-1 is the surface area of Sn-1,
.
The author's research was supported by a grant from NSFC. 相似文献
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