Igor Rivin
For a linear transformation A from Rn to Rn, we give sharp bounds for the average distortion of A, that is, the average value of log of the euclidean norm of Au over all unit vectors u. This is closely related to the results of the author's paper math.DS/0312048.
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We consider integral and series transformations, which are associated with Ramanujan's identities, involving various arithmetic functions and a ratio of products of Riemann's zeta functions of different arguments. Reciprocal inversion formulas are proved in a Banach space of functions whose Mellin's transforms are integrable over the vertical line Re s > 1. Examples of new transformations like Widder-Lambert and Kontorovich-Lebedev type are exhibited. Particular cases include familiar Lambert and Mobius transformations. Finally a class of equivalences of the Salem type to the Riemann hypothes…
In this paper, we show that the kernel function of Cauchy type for type $BC$ 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 usefulness of Riemann P-symbols in deriving identities involving the parametrized special function Hl is explored. Hl is the analytic local solution of the Heun equation, the canonical second-order differential equation on the Riemann sphere with four regular singular points. The identities discussed include ones coming from Moebius automorphisms and F-homotopies, and also quadratic and biquadratic transformations. The case when Hl is identical to a generalized hypergeometric function of 3F2 type is examined, and Pfaff and Euler transformations of 3F2(a1,a2,e+1;b1,e;x) are derived. They ex…
It is shown that the classical quadratic and cubic transformation identities satisfied by the hypergeometric function ${}_3F_2$ can be extended to include additional parameter pairs, which differ by integers. In the extended identities, which involve hypergeometric functions of arbitrarily high order, the added parameters are nonlinearly constrained: in the quadratic case, they are the negated roots of certain orthogonal polynomials of a discrete argument (dual Hahn and Racah ones). Specializations and applications of the extended identities are given, including an extension of Whipple's ident…
We obtain extensions of classical hypergeometric identities of Bailey and Whipple that transform nearly-poised and very-well-poised series to Saalschützian series, Saalschützian series to Saalschützian series, and very-well-poised and nearly-poised series to very-well-poised series. We employ a method in which summations and transformations of lower-order series are used to obtain transformations of higher-order series. By taking limits, we also obtain extensions of two classical quadratic transformations of Whipple and Bailey. Furthermore, we show how a number of other well-known results rega…
In this paper, we will obtain new algebraic transformations of the $_2F_1$-hypergeometric functions. The main novelty in our approach is the interpretation of identities among $_2F_1$-hypergeometric functions as identities among automorphic forms on different Shimura curves.
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 …
We present transformations relating the third transcendent of Painlevé with parameter sets located at the corners of the Weyl chamber for the symmetry group of the system, the affine Weyl group of the root system $ B^{(1)}_2 $, to those at the origin. This transformation entails a scaling of the independent variable, and implies additive identities for the canonical Hamiltonians and product identities for the $τ$-functions with these parameter sets.
We give several expansion and identities involving the Ramanujan function $A_q$ and the Stieltjes--Wigert polynomials. Special values of our idenitities give $m$-versions of some of the items on the Slater list of Rogers-Ramanujan type identities. We also study some bilateral extensions of certain transformations in the theory of basic hypergeometric functions.
The beta integral method proved itself as a simple nonetheless powerful method of generating hypergeometric identities at a fixed argument. In this paper we propose a generalization by substituting the beta density with a particular type of Meijer's G function. By application of our method to known transformation formulas we derive about forty hypergeometric identities, majority of which are believed to be new. We further apply some of these transformations to obtain several new summation formulas.
We derive formulas for the construction of all inequivalent Jacobian elliptic fibrations on the Kummer surface of two non-isogeneous elliptic curves from extremal rational elliptic surfaces by rational base transformations and quadratic twists. We then show that each such decomposition yields a description of the Picard-Fuchs system satisfied by the periods of the holomorphic two-form as either a tensor product of two Gauss' hypergeometric differential equations, an Appell hypergeometric system, or a GKZ differential system. As the answer must be independent of the fibration used, identities r…
Complex binomial theorem and pentagon identities
本文通过双曲超几何积分的极限研究复二项式定理与五边形恒等式，并推导出复伽马函数的傅里叶变换公式。
We study an equivalence class of iterated rational Darboux transformations applied on the harmonic oscillator, showing that many choices of state adding and state deleting transformations lead to the same transformed potential. As a by-product, we derive new identities between determinants whose entries are Hermite polynomials. These identities have a combinatorial interpretation in terms of Maya diagrams, partitions and Durfee rectangles, and serve to characterize the equivalence class of rational Darboux transformations. Since the determinants have different orders, we analyze the problem of…
This paper is an annotated list of transformation properties and identities satisfied by the four theta functions $θ_1$, $θ_2$, $θ_3$, $θ_4$ of one complex variable, presented in a ready-to-use form. An attempt is made to reveal a pattern behind various identities for the theta-functions. It is shown that all possible 3, 4 and 5-term identities of degree four emerge as algebraic consequences of the six fundamental bilinear 3-term identities connecting the theta-functions with modular parameters $τ$ and $2τ$.
Polynomial perturbations of Euler's and Clausen's identities
针对两个超几何级数乘积通常非超几何的问题，本文通过多项式扰动扩展欧拉变换与克劳森恒等式，提出Miller-Paris算子并给出简化证明，实现超几何级数乘积向单个超几何级数的转化。
In this article we prove a new elliptic hypergeometric integral identity. It previously appeared (as a conjecture) in articles by Rains, and Spiridonov and Vartanov. Moreover it gives a different proof of an identity in another article by Rains. We also give some basic hypergeometric and classical limits of this identity. The classical limit gives identities (some known, some new) between generalizations of the Selberg integral.
Spectral decomposition of dynamical equations using curl-eigenfunctions has been extensively used in fluid and plasma dynamics problems using their orthogonality and completeness properties for both linear and non-linear cases. Coefficients of such expansions are integrals over products of Bessel functions in problems involving cylindrical geometry. In this paper, certain identities involving products of two and three general solutions of Bessel's equation have been derived. Some of these identities have been useful in the study of Turner relaxation of annular magnetized plasma [S.K.H. Auluck,…
Kajihara obtained in 2004 a remarkable transformation formula connecting multiple basic hypergeometric series associated with $A$-type root systems of different ranks. By multiple principle specialisations of his formula, we deduce kernel identities for deformed Macdonald-Ruijsenaars (MR) and Noumi-Sano (NS) operators. The deformed MR operators were introduced by Sergeev and Veselov in the first order case and by Feigin and Silantyev in the higher order cases. As applications of our kernel identities, we prove that all of these operators pairwise commute and are simultaneously diagonalised by …
We obtain a family of functional identities satisfied by vector-valued functions of two variables and their geometric inversions. For this we introduce particular differential operators of arbitrary order attached to Gegenbauer polynomials. These differential operators are symmetry breaking for the pair of Lie groups $(SL(2,\mathbb C), SL(2,\mathbb R))$ that arise from conformal geometry.
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