Quasipower and hypergeometric series — construction and evaluation

It is proved that some power series converging very slowly in a neighbourhood of the point 1 can be transformed intoquasipower series. The latter converge faster but are more complicated because they contain some hypergeometric series₂ F ₁. Standard methods of the values evaluation for needed hypergeometric series with the aid of recurrence relations are not sufficiently efficient for some variable values. Therefore a new method, formally similar to Levin's transforms, is proposed. More generally, this is a method of approximative evaluating of such a solution of an inhomogeneous recurrence relation of order one which has some particular asymptotic properties.The efficacity of the proposed methods is analyzed in detail for Euler's dilogarithm. This is a typical function whose power series is approached with difficulties ifz1. In particular, its Padé approximants are sufficiently accurate only for, sayx[–1, 1/2]. Hermite-Padé approximation is more effective. Resulting irrational approximants generalize in some sense partial sums of the quasipower series introduced here.     

Blossoming and Hermite-Padé approximation for hypergeometric series

Numerical Algorithms - Based on the blossoming theory, in this work we develop a new method for deriving Hermite-Padé approximants of certain hypergeometric series. Its general principle...     

Kernel identities for van Diejen’s q-difference operators and transformation formulas for multiple basic hypergeometric series

The kernel function of Cauchy type for type BC is defined as a solution of linear q-difference equations. In this paper, we show that this kernel function intertwines the commuting family of van Diejen’s q-difference operators. This result gives rise to a transformation formula for certain multiple basic hypergeometric series of type BC. We also construct a new infinite family of commuting q-difference operators for which the Koornwinder polynomials are joint eigenfunctions.     

The zeros of the Weierstrass $$\wp$$–function and hypergeometric series

We express the zeros of the Weierstrass -function in terms of generalized hypergeometric functions. As an application of our main result we prove the transcendence of two specific hypergeometric functions at algebraic arguments in the unit disc. We also give a Saalschützian ₄ F ₃–evaluation. Research of W. Duke was supported in part by NSF Grant DMS-0355564. He wishes to acknowledge and thank the Forschungsinstitut für Mathematik of ETH Zürich for its hospitality and support.     

Bibasic analystic functions and discrete ‘bibasic’ hypergeometric series

This paper deals with a study of a class of functions called bibasis analytic functions. Using discrete powerz ⁽ⁿ⁾, discrete bibasic hypergeometric functions have been introduced.     

Relations among complementary and supplementary pairings of Saalschützian $${}_4F_3(1)$$ series

We investigate sums \(J(\vec {x})\) and \(L(\vec {x})\) of pairs of normalized Saalschützian \({}_4F_3(1)\) hypergeometric series, and develop a theory of relations among these J and L functions. The function \(L(\vec {x})\) has been studied extensively in the literature and has been shown to satisfy a number of two-term and three-term relations with respect to the variable \(\vec {x}\). More recent works have framed these relations in terms of Coxeter group actions on \(\vec {x}\) and have developed a similar theory of two-term and three-term relations for \(J(\vec {x})\). In this article, we derive “mixed” three-term relations, wherein any one of the L (respectively, J) functions arising in the above context may be expressed as a linear combination of two of the above J (respectively, L) functions. We show that, under the appropriate Coxeter group action, the resulting set of three-term relations (mixed and otherwise) among J and L functions partitions into eighteen orbits. We provide an explicit example of a relation from each orbit. We further classify the eighteen orbits into five types, with each type uniquely determined by the distances (under a certain natural metric) between the J and L functions in the relation. We show that the type of a relation dictates the complexity (in terms of both number of summands and number of factors in each summand) of the coefficients of the J and L functions therein.     

Common extension of bilateral series for Andrews’ q-Bailey and q-Gauss sums

By means of the modified Abel lemma on summation by parts, we establish a common bilateral series extension of the q-Bailey and q-Gauss sums discovered by Andrews (Duke Math. J. 40 (1973), 525–528).     

A note on Clebsch–Gordan integral,Fourier–Legendre expansions and closed form for hypergeometric series

The Ramanujan Journal - Let $$p\ge 3$$ be a prime number. In this article, we study the canonical expansion of the primitive $$p^n$$ -th root of unity $$\zeta _{p^n}$$ in p-adic...     

Secure collaborative supply chain planning and inverse optimization – The JELS model

It is a well-acknowledged fact that collaboration between different members of a supply chain yields a significant potential to increase overall supply chain performance. Sharing private information has been identified as prerequisite for collaboration and, at the same time, as one of its major obstacles. One potential avenue for overcoming this obstacle is Secure Multi-Party Computation (SMC). SMC is a cryptographic technique that enables the computation of any (well-defined) mathematical function by a number of parties without any party having to disclose its input to another party. In this paper, we show how SMC can be successfully employed to enable joint decision-making and benefit sharing in a simple supply chain setting. We develop secure protocols for implementing the well-known “Joint Economic Lot Size (JELS) Model” with benefit sharing in such a way that none of the parties involved has to disclose any private (cost and capacity) data. Thereupon, we show that although computation of the model’s outputs can be performed securely, the approach still faces practical limitations. These limitations are caused by the potential of “inverse optimization”, i.e., a party can infer another party’s private data from the output of a collaborative planning scheme even if the computation is performed in a secure fashion. We provide a detailed analysis of “inverse optimization” potentials and introduce the notion of “stochastic security”, a novel approach to assess the additional information a party may learn from joint computation and benefit sharing. Based on our definition of “stochastic security” we propose a stochastic benefit sharing rule, develop a secure protocol for this benefit sharing rule, and assess under which conditions stochastic benefit sharing can guarantee secure collaboration.     

The Menšov-Paley theorem and related results for double orthogonal series

клАссИЧЕскИЕ тЕОРЕМ ы Ф. РИссА, шлИ И МЕНьшО ВА (сМ., НАпРИМЕР, [3. стР. 102, 121 И 189]) РАспРОстРАНьУтсь, И пРИ ЁтОМ В ОБОБЩЕННОИ пОстАНОВкЕ МАРпИНкЕ ВИЧА И жИгМУНДА [1], с ОДНОМЕРНОгО НА ДВУМ ЕРНыИ слУЧАИ. пУсть {?_i(x)} (i=1, 2,...) И (ψ_k(y) (k=1, 2,...) — ДВЕ сИстЕМы ФУНкцИИ, ОРтО НОРМИРОВАННыЕ НА (НЕ О БьжАтЕльНО кОНЕЧНых) ИНтЕРВАлАх (А, ь) И (с, d), сООтВЕтстВЕННО. пР ЕДпОлАгАЕтсь, ЧтО ∥?_i¦|_v≦M_i И ψ_{k v}≦N_k Дль НЕкОтОРОгОv>2, гДЕМ _iИN _k кОНЕЧНы. ИжУ Ч АУтсь сВОИстВА схОДИ МОстИ ДВОИНых РьДОВ \(\mathop \Sigma \limits_{\iota = 1}^\infty \mathop \Sigma \limits_{\kappa = 1}^\infty c_{ik} \varphi _i (x)\psi _k (y).\) ОсНОВНОИ РЕжУльтАт с ОстОИт В слЕДУУЩЕМ Ут ВЕРжДЕНИИ. тЕОРЕМА 5. пУсть 2u ВыпО лНЕНО УслОВИЕ (3.2).ЕслИ Р ьД (3.3)схОДИтсь, тО РьД (1.16)схОДИтсь пОЧтИ ВсУД У пРИ пРОИжВОльНОИ пЕРЕстАНОВкЕ ЕгО ЧлЕ НОВ. ЕслИ (1.17) —пРОИжВОл ьНАь пЕРЕстАНОВкА РьДА (1.16),mО ВыпОлНЕНА ОцЕНкА (3.4),пР ИЧЕМ ВЕлИЧИНА Ag _q,v ^* жАВИсИт тОлькО От q И v. тЕОРЕМА 5 Дльv=∞ ьВльЕт сь ДВУМЕРНыМ АНАлОгО М тЕОРЕМы МЕНьшОВА—пЁлИ.     

A note on a theorem of W. Gaschütz and N. Itô

Given a finite groupG andp an odd prime number, we conclude thatO ^p(G)G isp-nilpotent when for every subgroupH ofG of orderp there exists a subgroupK ofG such thatG=HK andH permutes with every subgroup ofK.     

The evaluation of a weighted sum of Gauss hypergeometric functions and its connection with Galton–Watson processes

Lithuanian Mathematical Journal - We evaluate a weighted sum of Gauss hypergeometric functions for certain ranges of the argument, weights and parameters. We establish the domain of absolute...     

The Bishop–Phelps–Bollobás and approximate hyperplane series properties

We study the Bishop–Phelps–Bollobás property for operators between Banach spaces. Sufficient conditions are given for generalized direct sums of Banach spaces with respect to a uniformly monotone Banach sequence lattice to have the approximate hyperplane series property. This result implies that Bishop–Phelps–Bollobás theorem holds for operators from ${?}_1$ into such direct sums of Banach spaces. We also show that the direct sum of two spaces with the approximate hyperplane series property has such property whenever the norm of the direct sum is absolute.     

The convergence of padé approximants and the size of the power series coefficients

Let f be a power series ∑a_izⁱ with complex coefficients. The (n. n) Pade approximant to f is a rational function P/Q where P and Q are polynomials, Q(z) ? 0, of degree ≦ n such that f(z)Q(z)-P(z) = Az²ⁿ⁺¹ + higher degree terms. It is proved that if the coefficients a_i satisfy a certain growth condition, then a corresponding subsequence of the sequence of (n, n) Pade approximants converges to f in the region where the power series f converges, except on an exceptional set E having a certain Hausdorff measure 0. It is also proved that the result is best possible in the sense that we may have divergence on E. In particular,there exists an entire function f such that the sequence of (ny n) Pade approximants diverges everywhere (except at 0)     

Teichmüller curves in moduli space,Eisenstein series and an application to triangular billiards

Summary There exists a Teichmüller disc _n containing the Riemann surface ofy ²+x ⁿ =1, in the genus [n–1/2] Teichmüller space, such that the stabilizer of _n in the mapping class group has a fundamental domain of finite (Poincaré) volume in _n . Application is given to an asymptotic formula for the length spectrum of the billiard in isosceles triangles with angles (/n, /n,n–2/n) and to the uniform distribution of infinite billiard trajectories in the same triangles.

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Research supported by NSF-DMS-8521620     

The Möbius inversion formula for Fourier series applied to Bernoulli and Euler polynomials

Hurwitz found the Fourier expansion of the Bernoulli polynomials over a century ago. In general, Fourier analysis can be fruitfully employed to obtain properties of the Bernoulli polynomials and related functions in a simple manner. In addition, applying the technique of Möbius inversion from analytic number theory to Fourier expansions, we derive identities involving Bernoulli polynomials, Bernoulli numbers, and the Möbius function; this includes formulas for the Bernoulli polynomials at rational arguments. Finally, we show some asymptotic properties concerning the Bernoulli and Euler polynomials.     

The Poincaré series of the algebras of simultaneous invariants and covariants of two binary forms

Let ? _{d 1, d 2} and 𝒞 _{d 1, d 2} be the algebras of simultaneous invariants and simultaneous covariants of the two binary forms of degrees d ₁ and d ₂. Formulas for computation of the Poincaré series 𝒫? _{d 1, d 2} (z), 𝒫𝒞 _{d 1, d 2} (z) of the algebras are found. By using these formulas, we have computed the series for d ₁, d ₂ ≤ 20.     

The Littlewood-Offord problem in high dimensions and a conjecture of Frankl and Füredi

We give a new bound on the probability that the random sum ξ ₁ v ₁+…+ξ _n v _n belongs to a ball of fixed radius, where the ξ _i are i.i.d. Bernoulli random variables and the v _i are vectors in R ^d . As an application, we prove a conjecture of Frankl and Füredi (raised in 1988), which can be seen as the high dimensional version of the classical Littlewood-Offord-Erd?s theorem.