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1.
We prove certain identities between Bessel functions attached to irreducible unitary representations ofPGL
2(R) and Bessel functions attached to irreducible unitary representations of the double cover ofSL
2(R). These identities give a correspondence between such representations which turns out to be the Waldspurger correspondence.
In the process we prove several regularity theorems for Bessel distributions which appear in the relative trace formula. In
the heart of the proof lies a classical result of Weber and Hardy on a Fourier transform of classical Bessel functions. This
paper constitutes the local (real) spectral theory of the relative trace formula for the Waldspurger correspondence for which
the global part was developed by Jacquet.
Research of first author was partially supported by NSF grant DMS-0070762.
Research of second author was partially supported by NSF grant DMS-9729992 and DMS 9971003. 相似文献
2.
By means of Jacobi?s triple product identity and the t -coefficient method, we establish a general series expansion formula with five free parameters for the product of arbitrary two Jacobi theta functions. It embodies the triple, quintuple, sextuple and septuple theta function product identities and the generalized Schröter formula. As further applications, we also set up a series expansion formula for the product of three theta functions. It not only generalizes Ewell?s and Chen–Chen–Huang?s octuple product identities, but also contains three cubic theta function identities due to Farkas–Kra and Ramanujan respectively and the Macdonald identity for the root system A2 as special cases. In the meantime, many other new identities including a new short expression of the triple theta series of Andrews are also presented. 相似文献
3.
Sander Zwegers 《The Ramanujan Journal》2009,20(2):207-214
We consider the fifth order mock theta functions χ
0 and χ
1, defined by Ramanujan, and find identities for these functions, which relate them to indefinite theta functions. Similar
identities have been found by Andrews for the other fifth order mock theta functions and the seventh order functions. 相似文献
4.
Hiroshi Mizukawa 《Journal of Algebraic Combinatorics》2007,25(2):189-215
The pair of groups, symmetric group S
2n
and hyperoctohedral group H
n
, form a Gelfand pair. The characteristic map is a mapping from the graded algebra generated by the zonal spherical functions
of (S
2n
,H
n
) into the ring of symmetric functions. The images of the zonal spherical functions under this map are called the zonal polynomials.
A wreath product generalization of the Gelfand pair (S
2n
,H
n
) is discussed in this paper. Then a multi-partition versions of the theory is constructed. The multi-partition version of
zonal polynomials are products of zonal polynomials and Schur functions and are obtained from a characteristic map from the
graded Hecke algebra into a multipartition version of the ring of symmetric functions.
Dedicated to Professor Eiichi Bannai on his 60th birthday. 相似文献
5.
Raimundas Vidūnas 《Journal of Mathematical Analysis and Applications》2009,355(1):145-163
Univariate specializations of Appell's hypergeometric functions F1, F2, F3, F4 satisfy ordinary Fuchsian equations of order at most 4. In special cases, these differential equations are of order 2 and could be simple (pullback) transformations of Euler's differential equation for the Gauss hypergeometric function. The paper classifies these cases, and presents corresponding relations between univariate specializations of Appell's functions and univariate hypergeometric functions. The computational aspect and interesting identities are discussed. 相似文献
6.
Somjit Dutt 《Journal of Number Theory》2011,131(9):1547-1552
We prove two identities involving Dirichlet series, in the denominators of whose terms sums of two, three and four squares appear. These follow from two classical identities of Jacobi involving the four Jacobian Theta Functions θ1(z;q), θ2(z;q), θ3(z;q) and θ4(z;q), by the application of the Mellin transform. These results motivate the well-known correspondence between the set of the four Jacobian Theta Functions and the set of four classical zeta functions of which the Riemann Zeta Function is the third, and the Dirichlet Beta Function is the first. 相似文献
7.
Augustin Paulik 《Applicable analysis》2013,92(3):177-188
Two kinds of spaces of harmonic functions defined on m-dimensional domains are considered. In the first case, the spherical domain is a m-dimensional ball Br in the second case Br is replaced by Br1,r2:=Br2Br1. In both cases, several inner products are considered and Hilbert space properties are proved. The reproducing kernel functions of these spaces and hence some new representations for harmonic functions are also obtained 相似文献
8.
Stefan Stanimirovi? 《Applied mathematics and computation》2011,217(22):9122-9132
In this paper we use the Catalan matrix power as a tool for deriving identities involving Catalan numbers and hypergeometric functions. For that purpose, we extend earlier investigated relations between the Catalan matrix and the Pascal matrix by inserting the Catalan matrix power and particulary the squared Catalan matrix in those relations. We also pay attention to some relations between Catalan matrix powers of different degrees, which allows us to derive the simplification formula for hypergeometric function 3F2, as well as the simplification formula for the product of the Catalan number and the hypergeometric function 3F2. Some identities involving Catalan numbers, proved by the non-matrix approach, are also given. 相似文献
9.
We show here that a certain sequence of polynomials arising in the study of S2 m-quasi invariants satisfies a 3-term recursion. This leads to the discovery that these polynomials are closely related to the Bessel polynomials studied by Luc Favreau. This connection reveals a variety of combinatorial properties of the sequence of Baker–Akhiezer functions for S2. In particular we obtain in this manner their generating function and show that it is equivalent to several further identities satisfied by these functions. 相似文献
10.
Infinite Families of Exact Sums of Squares Formulas,Jacobi Elliptic Functions,Continued Fractions,and Schur Functions 总被引:2,自引:0,他引:2
In this paper we derive many infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi's 4 and 8 squares identities to 4n
2 or 4n(n + 1) squares, respectively, without using cusp forms. In fact, we similarly generalize to infinite families all of Jacobi's explicitly stated degree 2, 4, 6, 8 Lambert series expansions of classical theta functions. In addition, we extend Jacobi's special analysis of 2 squares, 2 triangles, 6 squares, 6 triangles to 12 squares, 12 triangles, 20 squares, 20 triangles, respectively. Our 24 squares identity leads to a different formula for Ramanujan's tau function (n), when n is odd. These results, depending on new expansions for powers of various products of classical theta functions, arise in the setting of Jacobi elliptic functions, associated continued fractions, regular C-fractions, Hankel or Turánian determinants, Fourier series, Lambert series, inclusion/exclusion, Laplace expansion formula for determinants, and Schur functions. The Schur function form of these infinite families of identities are analogous to the -function identities of Macdonald. Moreover, the powers 4n(n + 1), 2n
2 + n, 2n
2 – n that appear in Macdonald's work also arise at appropriate places in our analysis. A special case of our general methods yields a proof of the two Kac–Wakimoto conjectured identities involving representing a positive integer by sums of 4n
2 or 4n(n + 1) triangular numbers, respectively. Our 16 and 24 squares identities were originally obtained via multiple basic hypergeometric series, Gustafson's C
nonterminating 65 summation theorem, and Andrews' basic hypergeometric series proof of Jacobi's 2, 4, 6, and 8 squares identities. We have (elsewhere) applied symmetry and Schur function techniques to this original approach to prove the existence of similar infinite families of sums of squares identities for n
2 and n(n + 1) squares. Our sums of more than 8 squares identities are not the same as the formulas of Mathews (1895), Glaisher (1907), Sierpinski (1907), Uspensky (1913, 1925, 1928), Bulygin (1914, 1915), Ramanujan (1916), Mordell (1917, 1919), Hardy (1918, 1920), Bell (1919), Estermann (1936), Rankin (1945, 1962), Lomadze (1948), Walton (1949), Walfisz (1952), Ananda-Rau (1954), van der Pol (1954), Krätzel (1961, 1962), Bhaskaran (1969), Gundlach (1978), Kac and Wakimoto (1994), and, Liu (2001). We list these authors by the years their work appeared. 相似文献
11.
L. N. Lyakhov 《Mathematical Notes》2016,100(1-2):100-112
A formula relating the Radon transform of functions of spherical symmetries to the Radon–Kipriyanov transform Kγ for a naturalmulti-index γ is given. For an arbitrary multi-index γ, formulas for the representation of the Kγ-transform of a radial function as fractional integrals of Erdelyi–Kober integral type and of Riemann–Liouville integral type are proved. The corresponding inversion formulas are obtained. These results are used to study the inversion of the Radon–Kipriyanov transform of the generalized (generated by a generalized shift) spherical mean values of functions that belong to a weighted Lebesgue space and are even with respect to each of the weight variables. 相似文献
12.
A. V. Kashevarov 《Computational Mathematics and Mathematical Physics》2011,51(12):2137-2145
A spherical probe placed in a slowly moving collisional plasma with a large Debye length λD → ∞ is considered. The partial differential equation describing the electron concentration around the probe is reduced to
two ordinary differential equations, namely, to the equation for Coulomb spheroidal functions and Mathieu’s modified equation
with the parameter a of the latter related to the eigenvalue λ of the former by the relation a = λ + 1/4. It is shown that the solutions of Mathieu’s equation are Mathieu functions of half-integer order, which are expressed
as series in terms of spherical Bessel functions and series of products of Bessel functions. These Mathieu functions are numerically
constructed for Mathieu’s modified and usual equations. 相似文献
13.
We prove an identity for Hall–Littlewood symmetric functions labelled by the Lie algebra A2. Through specialization this yields a simple proof of the A2 Rogers–Ramanujan identities of Andrews, Schilling and the author. 相似文献
14.
We characterize genus g canonical curves by the vanishing of combinatorial products of g + 1 determinants of Brill-Noether matrices. This also implies the characterization of canonical curves in terms of (g?2)(g?3)/2 theta identities. A remarkable mechanism, based on a basis of H 0(K C ) expressed in terms of Szegö kernels, reduces such identities to a simple rank condition for matrices whose entries are logarithmic derivatives of theta functions. Such a basis, together with the Fay trisecant identity, also leads to the solution of the question of expressing the determinant of Brill-Noether matrices in terms of theta functions, without using the problematic Klein-Fay section σ. 相似文献
15.
N. G. Aharonyan 《Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)》2008,43(5):257-264
A family of identities primarily associated with isoperimetric inequalities for planar convex domains was discovered by Pleijel in 1956. We call these identities classical Pleijel identities. R. V. Ambartzumian gave combinatorial proof of these identities and pointed out that they can be applied to find chord length distribution functions for convex domains. In the classical Pleijel identities integration is over the measure in the space \(\mathbb{G}\) of lines which is invariant with respect to the all Euclidean motions. In the present paper they are considered for any locally-finite measure in the space \(\mathbb{G}\). These identities are applied to find the so-called orientation-dependent chord length distribution (or density) functions for bounded convex domains. 相似文献
16.
Rajesh K. Srivastava 《Israel Journal of Mathematics》2014,200(1):171-192
In this article, we review the Weyl correspondence of bigraded spherical harmonics and use it to extend the Hecke-Bochner identities for the spectral projections f × φ k n?1 for function f ∈ L p (? n ) with 1 ≤ p ≤ ∞. We prove that spheres are sets of injectivity for the twisted spherical means with real analytic weight. Then, we derive a real analytic expansion for the spectral projections f × φ k n?1 for function f ∈ L 2(? n ). Using this expansion we deduce that a complex cone can be a set of injectivity for the twisted spherical means. 相似文献
17.
S. Bhargava Chandrashekar Adiga D. D. Somashekara 《Proceedings Mathematical Sciences》1987,97(1-3):31-43
In this note we establish continued fraction developments for the ratios of the basic hypergeometric function2ϕ1(a,b;c;x) with several of its contiguous functions. We thus generalize and give a unified approach to establishing several
continued fraction identities including those of Srinivasa Ramanujan. 相似文献
18.
V. F. Babenko V. G. Doronin A. A. Ligun A. A. Shumeiko 《Ukrainian Mathematical Journal》2005,57(3):347-363
We obtain exact estimates for the approximation of functions defined on a sphere in the metrics of C and L
2 by linear methods of summation of Fourier series in spherical harmonics in the case where differential and difference properties
of these functions are defined in the space L
2.
__________
Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 3, pp. 291–304, March, 2005. 相似文献
19.
Rekha Srivastava 《Applied mathematics and computation》2011,218(3):1077-1083
The main object of this presentation is to show how some simple combinatorial identities can lead to several general families of combinatorial and other series identities as well as summation formulas associated with the Fox-Wright function pΨq and various related generalized hypergeometric functions. At least one of the hypergeometric summation formulas, which is derived here in this manner, has already found a remarkable application in producing several interesting generalizations of the Karlsson-Minton summation formula. We also consider a number of other combinatorial series identities and rational sums which were proven, in recent works, by using different methods and techniques. We show that much more general results can be derived by means of certain summation theorems for hypergeometric series. Relevant connections of the results presented here with those in the aforementioned investigations are also considered. 相似文献
20.
We present some identities and congruences for the general partition function p r (n). In particular, we deduce some known identities for Ramanujan’s tau function and find simple proofs of Ramanujan’s famous partition congruences for modulo 5 and 7. Our emphasis throughout this paper is to exhibit the use of Ramanujan’s theta functions to generate identities and congruences for general partition function. 相似文献