Bessel identities in the waldspurger correspondence over the real numbers

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.     

<Emphasis Type="Italic">q</Emphasis>-Hypergeometric Proofs of Polynomial Analogues of the Triple Product Identity,Lebesgue’s Identity and Euler’s Pentagonal Number Theorem

We present alternative, q-hypergeometric proofs of some polynomial analogues of classical q-series identities recently discovered by Alladi and Berkovich, and Berkovich and Garvan.Work supported by the Australian Research Council     

Andrews–Gordon type identities from combinations of Virasoro characters

For p∈{3,4} and all p′>p, with p′ coprime to p, we obtain fermionic expressions for the combination χ _1,s ^p,p′+q ^Δ χ _p−1,s ^p,p′ of Virasoro (W ₂) characters for various values of s, and particular choices of Δ. Equating these expressions with known product expressions, we obtain q-series identities which are akin to the Andrews–Gordon identities. For p=3, these identities were conjectured by Bytsko. For p=4, we obtain identities whose form is a variation on that of the p=3 cases. These identities appear to be new. The case (p,p′)=(3,14) is particularly interesting because it relates not only to W ₂, but also to W ₃ characters, and offers W ₃ analogues of the original Andrews–Gordon identities. Our fermionic expressions for these characters differ from those of Andrews et al. which involve Gaussian polynomials. BF is partially supported by grant number RFBR 05-01-01007, and OF by the Australian Research Council (ARC).     

Quantum immanants and higher Capelli identities

We consider some remarkable central elements of the universal enveloping algebraU(gl(n)) which we call quantum immanants. We express them in terms of generatorsE _ij ofU(gl(n)) and as differential operators on the space of matrices These expressions are a direct generalization of the classical Capelli identities. They result in many nontrivial properties of quantum immanants. The author is supported by the International Science Foundation and the Russian Fundamental Research Foundation.     

Padé approximations to the logarithm II: Identities, recurrences, and symbolic computation

Combinatorial identities that were needed in [25] are proved, mostly with C. Schneider’s computer algebra package Sigma. The form of the Padé approximation of the logarithm of arbitrary order is stated as a conjecture. 2000 Mathematics Subject Classification Primary—41A21, 05A19, 33F10 Supported by NRF-grant 2047226. Supported by NRF-grant 2053748. Supported by the Austrian Academy of Sciences, by the John Knopfmacher Research Centre for Applicable Analysis and Number Theory, and by the SFB-grant F1305 and the grant P16613-N12 of the Austrian FWF. Supported by NRF-grant 2053756.     

Identities of PI-Algebras Graded by a Finite Abelian Group

We consider associative PI-algebras over an algebraically closed field of zero characteristic graded by a finite abelian group G. It is proved that in this case the ideal of graded identities of a G-graded finitely generated PI-algebra coincides with the ideal of graded identities of some finite dimensional G-graded algebra. This implies that the ideal of G-graded identities of any (not necessary finitely generated) G-graded PI-algebra coincides with the ideal of G-graded identities of the Grassmann envelope of a finite dimensional (G × ?₂)-graded algebra, and is finitely generated as GT-ideal. Similar results take place for ideals of identities with automorphisms.     

Approximation and eigenvalue extrapolation of Stokes eigenvalue problem by nonconforming finite element methods

In this paper we analyze the stream function-vorticity-pressure method for the Stokes eigenvalue problem. Further, we obtain full order convergence rate of the eigenvalue approximations for the Stokes eigenvalue problem based on asymptotic error expansions for two nonconforming finite elements, Q ₁^rot and EQ ₁^rot. Using the technique of eigenvalue error expansion, the technique of integral identities and the extrapolation method, we can improve the accuracy of the eigenvalue approximations. This project is supported in part by the National Natural Science Foundation of China (10471103) and is subsidized by the National Basic Research Program of China under the grant 2005CB321701.     

The Bailey transform and false theta functions

An empirical exploration of five of Ramanujan's intriguing false theta function identities leads to unexpected instances of Bailey's transform which, in turn, lead to many new identities for both false and partial theta functions. 2000 Mathematics Subject Classification Primary—33D15, 11P83 Partially supported by National Science Foundation Grant DMS 0457003. Supported by the Australian Research Council.     

Some (2,n) combinatorial Stein groupoids

A Stein groupoid (quasigroup) is a groupoid (quasigroup) satisfying the identityx(xy)=yx. We show that, for certain two variable identities, the variety of Stein groupoids defined by any one of these identities has the properties that every groupoid in the variety is a quasigroup and that the free groupoid generated by two elements is of finite (small) order which we determine. These results provide characterizations of some Stein quasigroups of small order and we give some further characterizations involving other identities.     

Dendriform Analogues of Lie and Jordan Triple Systems

We use computer algebra to determine all the multilinear polynomial identities of degree ≤7 satisfied by the trilinear operations (a·b)·c and a·(b·c) in the free dendriform dialgebra, where a·b is the pre-Lie or the pre-Jordan product. For the pre-Lie triple products, we obtain one identity in degree 3, and three independent identities in degree 5, and we show that every identity in degree 7 follows from the identities of lower degree. For the pre-Jordan triple products, there are no identities in degree 3, five independent identities in degree 5, and ten independent irreducible identities in degree 7. Our methods involve linear algebra on large matrices over finite fields, and the representation theory of the symmetric group.     

Finite trigonometric sums and class numbers

Explicit evaluations of finite trigonometric sums arose in proving certain theta function identities of Ramanujan. In this paper, without any appeal to theta functions, several classes of finite trigonometric sums, including the aforementioned sums, are evaluated in closed form in terms of class numbers of imaginary quadratic fields.Mathematics Subject Classification (2000): Primary, 11L03; Secondary, 11R29, 11L10Research partially supported by grant MDA904-00-1-0015 from the National Security Agency.Revised version: 19 April 2004     

On normal and regular identities

We deal with two kinds of special identities: normal and regular, considered by Mel'nik, Ponka and other authors. We point out fundamental properties of these identities. Also in §2 we show that the lattice of all subvarieties of the variety defined by all normal identities of a given varietyV (called the normal part of a varietyV) is isomorphic to the direct product of the lattice L(V) and a two-element chain. This result (Theorem 3) is a strengthening of a result of Mel'nik [14]. Theorem 4 states that the word problem for free algebras of the variety defined by all normal identities ofV is solvable if and only if it is solvable forV, which is due to the property of regular identities, proved in [8]. In §3 we consider normal and regular consequences of a given set of identities. Theorem 6 shows that for a given varietyV, satisfying a nonregular absorption law, the lattice L(Mod(NR(V))) is isomorphic to the direct product of the lattice L(V) and a four-element lattice, with two atoms.Theorems in §4 collect some of results on the existence of a finite basis for normal and regular part of a given, finitely presented varietyV and of the finite basis property, as well, strengthening the result of Lakser, Padmanabhan and Platt [12].Results above can be applied for semigroup varieties.Presented by George Grätzer.     

Decomposition and Group Theoretic Characterization of Pairs of Inverse Relations of the Riordan Type

A new solution to Riordans problem of combinatorial identities classification is presented. An algebgraic characterization of pairs of inverse relations of the Riordan type is given. The use of the integral representation approach for generating new types of combinatorial identities is demonstrated. Supported in part by the National Sciences and Engineering Research Council of Canada on Grant NSERC-108343.Mathematics Subject Classifications (2000) combinatorics, algebra.     

Hypergeometric identities for 10 extended Ramanujan-type series

We prove, by the WZ-method, some hypergeometric identities which relate ten extended Ramanujan type series to simpler hypergeometric series. The identities we are going to prove are valid for all the values of a parameter a when they are convergent. Sometimes, even if they do not converge, they are valid if we consider these identities as limits.      

Polynomial Identities for Tangent Algebras of Monoassociative Loops

We introduce degree n Sabinin algebras, which are defined by the polynomial identities up to degree n in a Sabinin algebra. Degree 4 Sabinin algebras can be characterized by the polynomial identities satisfied by the commutator, associator, and two quaternators in the free nonassociative algebra. We consider these operations in a free power associative algebra and show that one of the quaternators is redundant. The resulting algebras provide the natural structure on the tangent space at the identity element of an analytic loop for which all local loops satisfy monoassociativity, a ² a ≡ aa ². These algebras are the next step beyond Lie, Malcev, and Bol algebras. We also present an identity of degree 5 which is satisfied by these three operations but which is not implied by the identities of lower degree.     

Polynomial identities for the ternary cyclic sum

We simplify the results of Bremner and Hentzel [J. Algebra 231 (2000) 387–405] on polynomial identities of degree 9 in two variables satisfied by the ternary cyclic sum [a, b, c] = abc + bca + cab in every totally associative ternary algebra. We also obtain new identities of degree 9 in three variables which do not follow from the identities in two variables. Our results depend on (i) the LLL algorithm for lattice basis reduction, and (ii) linearization operators in the group algebra of the symmetric group which permit efficient computation of the representation matrices for a non-linear identity. Our computational methods can be applied to polynomial identities for other algebraic structures.     

Generalizations of two identities of Guo and Yang

In this paper, we provide generalizations of two identities of Guo and Yang [2] for the q-binomial coe?cients. This approach allows us to derive new convolution identities for the complete and elementary symmetric functions. New identities involving q-binomial coe?cients are obtained as very special cases of these results. A new relationship between restricted partitions and restricted partitions into parts of two kinds is derived in this context.     

Identical relations for the forms with a Young symmetry

Let there be given a Young symmetrizere . Consider the space of multilinear forms obtained by actione . In the paper the characteristic property in terms of identities is found for a multilinear form to belong to this space. This property is analogous to the well-known identities for the cases of the spaces of the symmetric and skew-symmetric forms. In addition, a generalization of Garnir identities is obtained.     

Partition identities and the coin exchange problem

The number of partitions of n into parts divisible by a or b equals the number of partitions of n in which each part and each difference of two parts is expressible as a non-negative integer combination of a and b. This generalizes identities of MacMahon and Andrews. The analogous identities for three or more integers (in place of a,b) hold in certain cases.