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An infinite sequence of inequalities involving special values of the Riemann zeta function
Abstract
In this paper, we give an infinite sequence of inequalities involving the Riemann zeta function with even arguments and the Chebyshev-Stirling numbers of the first kind. This result is based on a recent connection between the Riemann zeta function and the complete homogeneous symmetric functions. An interesting asymptotic formula related to the n-th complete homogeneous symmetric function is conjectured in this context.
... The Riemann zeta function ζ(·), lambda function λ(·), and Dirichlet's eta function η(·) are defined as follows Zhu and Hua [1], Zhu [2], and Ge [3] obtained 1 − 2 −n ζ(n) is decreasing. Using the special property of the complete homogeneous symmetric function, Merca [4] came up with the property of some combination constancy for a finite number of such functions ζ(n). Alzer and Kwong [5] proved that η (s) is strictly increasing on (0, ∞). ...
In the paper, the author discusses the monotonic properties of the functions involving multiple logarithm function by the monotone form of the L’Hôspitil rule and its improvement, and obtains the monotonic properties of a generalized function of lambda function and Dirichlet’s eta function.
... The complete homogeneous symmetric polynomial with n variables x 1 , . . . , x n and degree d ∈ N is defined as follows: Recently, several applications of such polynomials into combinatorics about Riemann zeta function are mentioned [12]. ...
This article deals with the positivity of a nice family of symmetric polynomials, namely complete homogeneous symmetric polynomials. We are able to give a positive answer to a question arising in Tao (https://terrytao.wordpress.com/2017/08/06/schur-convexity-and-positive-definiteness-of-the-even-degree-complete-homogeneous-symmetric-polynomials/, 2017). Our strategy follows two different ideas, one of them based on a Schur-convexity argument and the other one uses a method with divided differences. Several Newton’s type inequalities are also discussed.
... Rights reserved. As an important application of Theorems 1.1 and 1.2 we prove the following result, which was recently conjectured by the second author [25,Conjecture 2]. ...
In this paper, we give asymptotic formulas that combine the Euler-Riemann zeta function and the Chebyshev-Stirling numbers of the first kind. These results allow us to prove an asymptotic formula related to the $n$th complete homogeneous symmetric function, which was recently conjectured by the second author:
$$h_{n}\left(1,\left( \frac{k}{k+1}\right)^2 ,\left( \frac{k}{k+2} \right)^2 ,\ldots \right) \sim \binom{2k}{k}\quad\text{as}\quad n\to\infty.$$A direct proof of this asymptotic formula, due to Gerg\H{o} Nemes, is provided in the appendix.
In the paper, the author obtains some new bounds for the ratio of two adjacent even-indexed Bernoulli numbers, solves Qi’s conjecture on the related topic, and shows a tighter lower bound for this ratio. Then we pose some conjectures on related topics.
The asymptotic behaviour of the Chebyshev–Stirling numbers of the second kind, a special case of the Jacobi–Stirling numbers, has been established in a recent paper by Gawronski, Littlejohn and Neuschel. In this paper, we provide an asymptotic formula for the Chebyshev–Stirling numbers of the first kind. New recurrence relations for the Euler–Riemann zeta function (Formula presented.) are derived in this context.
Accurate approximations in terms of the Chebyshev–Stirling numbers of the first kind are established in the paper for the cardinal sine function. Similar results are presented for the hyperbolic cardinal sine function.
A finite discrete convolution involving the Jacobi-Stirling numbers of both kinds is expressed in this paper in terms of the Bernoulli polynomials.
The Jacobi–Stirling numbers of both kinds are specializations of elementary and complete homogeneous symmetric functions. We use this fact to discover and prove some algebraic identities involving Jacobi–Stirling numbers. The Legendre–Stirling numbers are very special cases of the Jacobi–Stirling numbers. New connections between the Legendre–Stirling numbers and the central factorial numbers of odd indices are presented.
For the classical Stirling numbers of the second kind, asymptotic formulae are derived in terms of a local central limit theorem. The underlying probabilistic approach also applies to the Chebyshev-Stirling numbers, a special case of the Jacobi-Stirling numbers. Essential features are uniformity properties and the fact that the leading terms of the asymptotics are given explicitly and they contain elementary expressions only. Thereby supplements of the asymptotic analysis of these numbers are established.
The Legendre-Stirling numbers were discovered in 2002 as a result of a problem in- volving the spectral theory of powers of the classical second-order Legendre dierential expression. Speci…cally, these numbers are the coe¢ cients of integral composite powers of the Legendre expres- sion in Lagrangian symmetric form. Quite remarkably, they share many similar properties with the classical Stirling numbers of the second kind which, as shown in (9), are the coe¢ cients of integral powers of the Laguerre dierential expression. An open question, regarding the Legendre-Stirling numbers, has been to obtain a combinatorial interpretation of these numbers. In this paper, we provide such an interpretation.
In this paper we give a convolution identity for complete and elementary symmetric functions. This result can be used to prove and discover some combinatorial identities involving r-Stirling numbers, r-Whitney numbers and q-binomial coefficients. As a corollary we derive a generalization of the quantum Vandermonde’s convolution identity.
The Legendre–Stirling numbers are the coefficients in the integral Lagrangian symmetric powers of the classical Legendre second-order differential expression. In many ways, these numbers mimic the classical Stirling numbers of the second kind which play a similar role in the integral powers of the classical second-order Laguerre differential expression. In a recent paper, Andrews and Littlejohn gave a combinatorial interpretation of the Legendre–Stirling numbers. In this paper, we establish several properties of the Legendre–Stirling numbers; as with the Stirling numbers of the second kind, they have interesting generating functions and recurrence relations. Moreover, there are some surprising and intriguing results relating these numbers to some classical results in algebraic number theory.
The Jacobi-Stirling numbers of the first and second kinds were introduced in 2006 in the spectral theory and are polynomial refinements of the Legendre-Stirling numbers. Andrews and Littlejohn have recently given a combinatorial interpretation for the second kind of the latter numbers. Noticing that these numbers are very similar to the classical central factorial numbers, we give combinatorial interpretations for the Jacobi-Stirling numbers of both kinds, which provide a unified treatment of the combinatorial theories for the two previous sequences and also for the Stirling numbers of both kinds. Comment: 15 pages
We show how the Binomial Theorem can be used to continue the Riemann Zeta Function to the left hand half-plane. This method yields the explicit values of the function at non-positive integers in terms of the Bernoulli numbers. Comment: Extra references added
For the Legendre-Stirling numbers of the second kind asymptotic formulae are
derived in terms of a local central limit theorem. Thereby, supplements of the
recently published asymptotic analysis of the Chebyshev-Stirling numbers are
established. Moreover, we provide results on the asymptotic normality and
unimodality for modified Legendre-Stirling numbers.
We develop the left-definite analysis associated with the self-adjoint
Jacobi operator , generated from the classical second-order Jacobi
differential expressionin the Hilbert space , where
w[alpha],[beta](t)=(1-t)[alpha](1+t)[beta], that has the Jacobi
polynomials as eigenfunctions; here, [alpha],[beta]>-1 and k is a
fixed, non-negative constant. More specifically, for each , we
explicitly determine the unique left-definite Hilbert-Sobolev space and
the corresponding unique left-definite self-adjoint operator in
associated with the pair . The Jacobi polynomials form a complete
orthogonal set in each left-definite space and are the eigenfunctions of
each . Moreover, in this paper, we explicitly determine the domain of
each as well as each integral power of . The key to determining these
spaces and operators is in finding the explicit Lagrangian symmetric
form of the integral composite powers of l[alpha],[beta],k[[middle
dot]]. In turn, the key to determining these powers is a double sequence
of numbers which we introduce in this paper as the Jacobi-Stirling
numbers. Some properties of these numbers, which in some ways behave
like the classical Stirling numbers of the second kind, are established
including a remarkable, and yet somewhat mysterious, identity involving
these numbers and the eigenvalues of .
We investigate the diagonal generating function of the Jacobi–Stirling numbers of the second kind JS(n+k,n;z) by generalizing the analogous results for the Stirling and Legendre–Stirling numbers. More precisely, letting JS(n+k,n;z)=pk,0(n)+pk,1(n)z+⋯+pk,k(n)zk, we show that (1−t)3k−i+1∑n≥0pk,i(n)tn(1−t)3k−i+1∑n≥0pk,i(n)tn is a polynomial in tt with nonnegative integral coefficients and provide combinatorial interpretations of the coefficients by using Stanley’s theory of PP-partitions.
The Jacobi–Stirling numbers of the first and second kinds were first introduced in Everitt et al. (2007) [8] and they are a generalization of the Legendre–Stirling numbers. Quite remarkably, they share many similar properties with the classical Stirling numbers. In this paper we study total positivity properties of these numbers. In particular, we prove that the matrix whose entries are the Jacobi–Stirling numbers is totally positive and that each row and each column is a Pólya frequency sequence, except for the columns with (unsigned) numbers of the first kind.
The Jacobi-Stirling numbers were discovered as a result of a problem
involving the spectral theory of powers of the classical second-order Jacobi
differential expression. Specifically, these numbers are the coefficients of
integral composite powers of the Jacobi expression in Lagrangian symmetric
form. Quite remarkably, they share many properties with the classical Stirling
numbers of the second kind which, as shown in LW, are the coefficients of
integral powers of the Laguerre differential expression. In this paper, we
establish several properties of the Jacobi-Stirling numbers and its companions
including combinatorial interpretations thereby extending and supplementing
known contributions to the literature of Andrews-Littlejohn,
Andrews-Gawronski-Littlejohn, Egge, Gelineau-Zeng, and Mongelli.
The Jacobi-Stirling numbers and the Legendre-Stirling numbers of the first and second kind were first introduced by Everitt et al. (2002) and (2007) in the spectral theory. In this paper we note that Jacobi-Stirling numbers and Legendre-Stirling numbers are specializations of elementary and complete symmetric functions. We then study combinatorial interpretations of this specialization and obtain new combinatorial interpretations of the Jacobi-Stirling and Legendre-Stirling numbers. Keywords: Jacobi-Stirling numbers, Legendre-Stirling numbers, symmetric functions, combinatorial interpretations. 1
Elementary evaluation of ζ (2n)
- B C Berndt
B. C. Berndt, Elementary evaluation of ζ (2n), Math. Magazine, 48, 3 (1975), 148-154.
e-mail: mircea.merca@profinfo.edu.ro Corresponding Author: Mircea Merca
- Mircea Merca
Mircea Merca, Academy of Romanian Scientists,
Splaiul Independentei 54, Bucharest, 050094 Romania
e-mail: mircea.merca@profinfo.edu.ro
Corresponding Author: Mircea Merca