István Mező
We shall show that the sum of the series formed by the so-called hyperharmonic numbers can be expressed in terms of the Riemann zeta function. More exactly, we give summation formula for the general hyperharmonic series.
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In 1990, Spieß\, gave some identities of harmonic numbers including the types of $\sum_{\ell=1}^n\ell^k H_\ell$, $\sum_{\ell=1}^n\ell^k H_{n-\ell}$ and $\sum_{\ell=1}^n\ell^k H_\ell H_{n-\ell}$. In this paper, we derive several formulas of hyperharmonic numbers including $\sum_{\ell=0}^{n} {\ell}^{p} h_{\ell}^{(r)} h_{n-\ell}^{(s)}$ and $\sum_{\ell=0}^n \ell^{p}(h_{\ell}^{(r)})^{2}$. Some more formulas of generalized hyperharmonic numbers are also shown.
We evaluate in closed form several series involving products of Cauchy numbers with other special numbers (harmonic, skew-harmonic, hyperharmonic, and central binomial). Similar results are obtained with series involving Stirling numbers of the first kind. We focus on several particular cases which give new closed forms for Euler sums of hyperharmonic numbers and products of hyperharmonic and harmonic numbers.
This paper presents a number of identities for Dirichlet series and series with Stirling numbers of the first kind. As coefficients for the Dirichlet series we use Cauchy numbers of the first and second kinds, hyperharmonic numbers, derangement numbers, binomial coefficients, central binomial coefficients, and Catalan numbers.
According to the method of series rearrangement, we establish two families of summation formulae for $q$-Watson type $_4ϕ_3$-series.
By using contiguous relations for basic hypergeometric series, we give simple proofs of Bailey's $_4ϕ_3$ summation, Carlitz's $_5ϕ_4$ summation, Sears' $_3ϕ_2$ to $_5ϕ_4$ transformation, Sears' ${}_4ϕ_3$ transformations, Chen's bibasic summation, Gasper's split poised $_{10}ϕ_9$ transformation, Chu's bibasic symmetric transformation. Along the same line, finite forms of Sylvester's identity, Jacobi's triple product identity, and Kang's identity are also obtained.
In terms of the difference operators, we establish several curious transformation and summation formulas for basic hypergeometric series. When the parameters are specified, they produce $q$-analogues of Ramanujan's three series for 1/$π$ and other eleven nice $π$-formulas. Meanwhile, $q$-analogues of three beautiful series for $ζ(3)$ are also given in the same way.
A $q$-Exponential Operator Based on the Derivative of Order 1 and Summation of Bilateral Basic Hypergeometric Series
本文提出一种基于一阶q^{±1}导数的q-指数算子，用于推导多种双边基本超几何级数的求和公式。
We show that several terminating summation and transformation formulas for basic hypergeometric series can be proved in a straightforward way. Along the same line, new finite forms of Jacobi's triple product identity and Watson's quintuple product identity are also proved.
Summation formulas for Fox-Wright function
利用反演技术和已知超几何级数恒等式，推导出Fox-Wright函数的求和公式，并得到新的超几何级数恒等式。
By some hypergeometric summation theorems, the authors establish a series of new infinite summation formulas involving generalized harmonic numbers related to Riemann-Zeta function, with three different patterns.
With the help of the partial derivative operator and several summation formulas for hypergeometric series, we find three double series for $π$. In terms of the operator just stated and several summation formulas for basic hypergeometric series, we also establish $q$-analogues of these double series.
Two parametric q-supercongruences from a summation formula for q-series
本文利用q级数求和公式与创造性显微镜方法，建立了两个以分圆多项式三次幂为模的参数化q-超同余式，并在q→1时推导出一个具体的模p^3同余结论。
We present a systematic method for proving nonterminating basic hypergeometric identities. Assume that $k$ is the summation index. By setting a parameter $x$ to $xq^n$, we may find a recurrence relation of the summation by using the $q$-Zeilberger algorithm. This method applies to almost all nonterminating basic hypergeometric summation formulas in the book of Gasper and Rahman. Furthermore, by comparing the recursions and the limit values, we may verify many classical transformation formulas, including the Sears-Carlitz transformation, transformations of the very-well-poised $_8ϕ_7$ series, t…
In 2017, He [Proc. Amer. Math. Soc. 145 (2017), 501--508] established two spuercongruences on truncated hypergeometric series and further proposed two related conjectures. Subsequently, Liu [Results Math. 72 (2017), 2057--2066] extended He's formulas and confirmed the second conjecture. However, the first conjecture is still open up to now. With the help of the creative microscoping method and the Chinese remainder theorem for coprime polynomials, we derive several $q$-supercongruences modulo the fourth and fifth powers of a cyclotomic polynomial from Gasper and Rahman's summation formula for …
Using matrix inversion and determinant evaluation techniques we prove several summation and transformation formulas for terminating, balanced, very-well-poised, elliptic hypergeometric series.
In terms of the derivative operator and Watson-type $_3F_2$-series identities, three families of summation formulae involving generalized harmonic numbers are established.
Using some properties of the gamma function and the well-known Gauss summation formula for the classical hypergeometric series, we prove a four-parameter series expansion formula, which can produce infinitely many Ramanujan type series for $1/π$.
By means of the derivative operator and Whipple-type $_3F_2$-series identities, two families of summation formulae involving generalized harmonic numbers are established.
In this paper we introduce the so-called truncated very-well-poised $_6ψ_6$ series and set up an explicit recurrence relation for it by means of the classical Abel lemma on summation by parts. This new recurrence relation implies an elementary proof of Bailey's well-known $_6ψ_6$ summation formula.
Several new identities for elliptic hypergeometric series are proved. Remarkably, some of these are elliptic analogues of identities for basic hypergeometric series that are balanced but not very-well-poised.
With the help of a summation of basic hypergeometric series, the creative microscoping method recently introduced by Guo and Zudilin, and the Chinese remainder theorem for coprime polynomials, we find some new $q$-supercongruences. Especially, we give a $q$-analogue of a formula due to Liu [J. Math. Anal. Appl. 497 (2021), Art.~124915].
We generalize a terminating summation formula to a unilateral nonterminating, and further, a bilateral summation formula by a property of analytic functions. The unilateral one is proved to be a $q$-analogue of a $_4F_3$-summation formula. And, an identity unifying Jacobi's triple product identity and the quintuple product identity is obtained as a special case of the bilateral one.
We show that several classical bilateral summation and transformation formulas have semi-finite forms. We obtain these semi-finite forms from unilateral summation and transformation formulas. Our method can be applied to derive Ramanujan's $_1ψ_1$ summation, Bailey's $_2ψ_2$ transformations, and Bailey's $_6ψ_6$ summation.
We give new proofs for certain bilateral basic hypergeometric summation formulas using the symmetries of the corresponding series. In particular, we present a proof for Bailey's $_3ψ_3$ summation formula as an application. We also prove a multiple series analogue of this identity by considering hyperoctahedral group symmetries of higher ranks.
From two q-summation formulas we deduce certain series expansion formulas involving the q-gamma function. With these formulas we can give q-analogues of series expansions for certain constants.
We use elliptic Taylor series expansions and interpolation to deduce a number of summations for elliptic hypergeometric series. We extend to the well-poised elliptic case results that in the $q$-case have previously been obtained by Cooper and by Ismail and Stanton. We also provide identities involving S. Bhargava's cubic theta functions.
Using multiple q-integrals and a determinant evaluation, we establish a multivariable extension of Bailey's nonterminating 10-phi-9 transformation. From this result, we deduce new multivariable terminating 10-phi-9 transformations, 8-phi-7 summations and other identities. We also use similar methods to derive new multivariable 1-psi-1 summations. Some of our results are extended to the case of elliptic hypergeometric series.
In terms of the derivative operator and three hypergeometric series identities, several interesting summation formulas involving generalized harmonic numbers are established.
We give elementary derivations of some classical summation formulae for bilateral (basic) hypergeometric series. In particular, we apply Gauss' 2-F-1 summation and elementary series manipulations to give a simple proof of Dougall's 2-H-2 summation. Similarly, we apply Rogers' nonterminating 6-phi-5 summation and elementary series manipulations to give a simple proof of Bailey's very-well-poised 6-psi-6 summation. Our method of proof extends M. Jackson's first elementary proof of Ramanujan's 1-psi-1 summation.
In terms of Sear's transformation formula for $_4ϕ_3$-series, we give new proofs of a summation formula for ${_4ϕ_3}$-series due to Andrews [2] and another summation formula for${_4ϕ_3}$-series conjectured in the same paper. Meanwhile, other several related results are also derived.
In this article, we derive some identities for multilateral basic hypergeometric series associated to the root system A_n. First, we apply Ismail's argument to an A_n q-binomial theorem of Milne and derive a new A_n generalization of Ramanujan's 1-psi-1 summation theorem. From this new A_n 1-psi-1 summation and from an A_n 1-psi-1 summation of Gustafson we deduce two lemmas for deriving simple A_n generalizations of bilateral basic hypergeometric series identities. These lemmas are closely related to the Macdonald identities for A_n. As samples for possible applications of these lemmas, we pro…
The Euler-MacLaurin summation formula relates a sum of a function to a corresponding integral, with a remainder term. The remainder term has an asymptotic expansion, and for a typical analytic function, it is a divergent (Gevrey-1) series. Under some decay assumptions of the function in a half-plane (resp. in the vertical strip containing the summation interval), Hardy (resp. Abel-Plana) prove that the asymptotic expansion is a Borel summable series, and give an exact Euler-MacLaurin summation formula. Using a mild resurgence hypothesis for the function to be summed, we give a Borel summable t…
In [Electron. J. Combin. 10 (2003), #R10], the author presented a new basic hypergeometric matrix inverse with applications to bilateral basic hypergeometric series. This matrix inversion result was directly extracted from an instance of Bailey's very-well-poised 6-psi-6 summation theorem, and involves two infinite matrices which are not lower-triangular. The present paper features three different multivariable generalizations of the above result. These are extracted from Gustafson's A_r and C_r extensions and of the author's recent A_r extension of Bailey's 6-psi-6 summation formula. By combi…
We obtain a formula which reduces the evaluation of a $_2ψ_2$ series to two $_2ϕ_1$ series. In some sense, this identity may be considered as a companion of Slater's formulas. We also find that a two-term ${}_2ψ_2$ summation formula due to Slater can be derived from a unilateral summation formula of Andrews by bilateral extension and parameter augmentation.
The Fibonacci number is the residue of a rational function, from which follows that Fibonacci number summation identities can be derived with the integral representation method, a method also used to derive combinatorial identities. A number of weighted generalized Fibonacci number summation identities are derived this way. From these identities some infinite series, generating functions and convolution identities are obtained. In addition, some weighted generalized Fibonacci number summation identities with binomial coefficients are derived. Many examples of both types of summation identities…
Analytic summation of series involving higher-order derivatives of Chebyshev polynomials of the second kind and their applications to convolved linear recurrent sequences
本文通过解析求和确定Chebyshev二类多项式高阶导数级数的收敛有理函数，并推导出Fibonacci、Lucas和Pell数及其卷积的组合恒等式。
We present a new matrix inverse with applications in the theory of bilateral basic hypergeometric series. Our matrix inversion result is directly extracted from an instance of Bailey's very-well-poised 6-psi-6 summation theorem, and involves two infinite matrices which are not lower-triangular. We combine our bilateral matrix inverse with known basic hypergeometric summation theorems to derive, via inverse relations, several new identities for bilateral basic hypergeometric series.
Using Krattenthaler's operator method, we give a new proof of Warnaar's recent elliptic extension of Krattenthaler's matrix inversion. Further, using a theta function identity closely related to Warnaar's inversion, we derive summation and transformation formulas for elliptic hypergeometric series of Karlsson-Minton-type. A special case yields a particular summation that was used by Warnaar to derive quadratic, cubic and quartic transformations for elliptic hypergeometric series. Starting from another theta function identity, we derive yet different summation and transformation formulas for el…
A family of orthogonal functions on the unit circle and a new multilateral matrix inverse
本文利用Bailey求和证明了特定级数构成单位圆上的正交函数族，并从Dougall求和提取出新的双边矩阵逆，结合Pfaff-Saalschütz求和推导了 $_3H_3$ 级数求和公式及其多变量扩展。
We give elementary derivations of several classical and some new summation and transformation formulae for bilateral basic hypergeometric series. For purpose of motivation, we review our previous simple proof ("A simple proof of Bailey's very-well-poised 6-psi-6 summation", Proc. Amer. Math. Soc., to appear) of Bailey's very-well-poised 6-psi-6 summation. Using a similar but different method, we now give elementary derivations of some transformations for bilateral basic hypergeometric series. In particular, these include M. Jackson's very-well-poised 8-psi-8 transformation, a very-well-poised …
Factorization of Basic Hypergeometric Series
本文研究基本超几何级数的因式分解问题，重点分析一般 $_2ψ_2$ 级数，并建立与根系上基本超几何级数理论的联系，同时给出若干经典求和与变换公式的新证明。
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