Mixed moments of L-functions

In this paper we will study the main term of mixed moments of L-functions associated to holomorphic cusp forms of integral weight on congruence subgroups of odd and squarefree level. 2000 Mathematics Subject Classification Primary—11F67, 11M41, 11F30     

Imaginary quadratic fields with noncyclic ideal class groups

Let g be an odd positive integer and X be a positive real number. We shall show that for any ∊ > 0, the number of imaginary quadratic fields with discriminant ≥ − X and ideal class group having a subgroup isomorphic to ℤ/g ℤ× ℤ/g ℤ is ≫ X^1/g−∊. 2000 Mathematics Subject Classification Primary—11R11, 11R29 This work was supported by KRF-R08-2003-000-10243-0 and partially by KRF-2002-003-C00001.     

Preface

We will interpret a partial theta identity in Ramanujan’s Lost Notebook as a weighted partition theorem involving partitions into distinct parts with smallest part odd. A special case of this yields a new result on the parity of the number of parts in such partitions, comparable to Euler’s pentagonal numbers theorem. We will provide simple and novel proofs of the weighted partition theorem and the special case. Our proof leads to a companion to Ramanujan’s partial theta identity which we will explain combinatorially.     

An inner product relation on Saito-Kurokawa lifts

Let f be a newform of weight 2k−2 and level M with M an odd square-free integer. Via the Saito-Kurokawa correspondence there is associated to f a Siegel newform F _f of weight k and level M. In this paper we provide a formula relating the Petersson products {F _f , F _f } and {f, f}. We use this result to give a new proof of a special case of a well-known result of Shimura on the algebraicity of a special value of a Rankin convolution L-function. 2000 Mathematics Subject ClassificationPrimary—11F32; Secondary—11F46, 11F67     

A Note on Partitions into Distinct Parts and Odd Parts

Bousquet-Mélou and Eriksson showed that the number of partitions of n into distinct parts whose alternating sum is k is equal to the number of partitions of n into k odd parts, which is a refinement of a well-known result by Euler. We give a different graphical interpretation of the bijection by Sylvester on partitions into distinct parts and partitions into odd parts, and show that the bijection implies the above statement.     

A note on slant submanifolds of nearly trans-Sasakian manifolds

The geometry of slant submanifolds of a nearly trans-Sasakian manifold is studied when the tensor field Q is parallel. It is proved that Q is not parallel on the submanifold unless it is anti-invariant and thus the result of [CABRERIZO, J. L.—CARRIAZO, A.—FERNANDEZ, L. M.—FERNANDEZ, M.: Slant submanifolds in Sasakian manifolds, Glasg. Math. J. 42 (2000), 125–138] and [GUPTA, R. S.—KHURSHEED HAIDER, S. M.—SHARFUDIN, A.: Slant submanifolds of a trans-Sasakian manifold, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 47 (2004), 45–57] are generalized.     

On a partition identity of Lehmer

Euler's identity equates the number of partitions of any non-negative integer n into odd parts and the number of partitions of n into distinct parts. Beck conjectured and Andrews proved the following companion to Euler's identity: the excess of the number of parts in all partitions of n into odd parts over the number of parts in all partitions of n into distinct parts equals the number of partitions of n with exactly one even part (possibly repeated). Beck's original conjecture was followed by generalizations and so-called “Beck-type” companions to other identities.In this paper, we establish a collection of Beck-type companion identities to the following result mentioned by Lehmer at the 1974 International Congress of Mathematicians: the excess of the number of partitions of n with an even number of even parts over the number of partitions of n with an odd number of even parts equals the number of partitions of n into distinct, odd parts. We also establish various generalizations of Lehmer's identity, and prove related Beck-type companion identities. We use both analytic and combinatorial methods in our proofs.     

Nonexistence of Certain Spherical Designs of Odd Strengths and Cardinalities

A spherical τ -design on S ^n-1 is a finite set such that, for all polynomials f of degree at most τ , the average of f over the set is equal to the average of f over the sphere S ^n-1 . In this paper we obtain some necessary conditions for the existence of designs of odd strengths and cardinalities. This gives nonexistence results in many cases. Asymptotically, we derive a bound which is better than the corresponding estimation ensured by the Delsarte—Goethals—Seidel bound. We consider in detail the strengths τ =3 and τ =5 and obtain further nonexistence results in these cases. When the nonexistence argument does not work, we obtain bounds on the minimum distance of such designs. Received January 30, 1997, and in revised form November 29, 1997.     

Some argument inequalities for certain analytic functions

For analytic functions f(z) in the open unit disk U and convex functions g(z) in U, Nunokawa et al. [NUNOKAWA, M.—OWA, S.—NISHIWAKI, J.—KUROKI, K.—HAYAMI, T: Differential subordination and argumental property, Comput. Math. Appl. 56 (2008), 2733–2736] have proved one theorem which is a generalization of the result [POMMERENKE, CH.: On close-toconvex analytic functions, Trans. Amer. Math. Soc. 114 (1965), 176–186]. The object of the present paper is to generalize the theorem due to Nunokawa et al..     

Borwein's conjecture on average over arithmetic progressions

We provide asymptotic formulas for sums over arithmetic progressions of coefficients of products of the form

Balanced partitions

A famous theorem of Euler asserts that there are as many partitions of n into distinct parts as there are partitions into odd parts. We begin by establishing a less well-known companion result, which states that both of these quantities are equal to the number of partitions of n into even parts along with exactly one triangular part. We then introduce the characteristic of a partition, which is determined in a simple way by the placement of odd parts within the list of all parts. This leads to a refinement of the aforementioned result in the form of a new type of partition identity involving characteristic, distinct parts, even parts, and triangular numbers. Our primary purpose is to present a bijective proof of the central instance of this new type of identity, which concerns balanced partitions—partitions in which odd parts occupy as many even as odd positions within the list of all parts. The bijection is accomplished by means of a construction that converts balanced partitions of 2n into unrestricted partitions of n via a pairing of the squares in the Young tableau.     

Bilinear Generating Functions for Orthogonal Polynomials

Using realizations of the positive discrete series representations of the Lie algebra su(1,1) in terms of Meixner—Pollaczek polynomials, the action of su(1,1) on Poisson kernels of these polynomials is considered. In the tensor product of two such representations, two sets of eigenfunctions of a certain operator can be considered and they are shown to be related through continuous Hahn polynomials. As a result, a bilinear generating function for continuous Hahn polynomials is obtained involving the Poisson kernel of Meixner—Pollaczek polynomials; this result is also known as the Burchnall—Chaundy formula. For the positive discrete series representations of the quantized universal enveloping algebra U _q (su(1,1)) a similar analysis is performed and leads to a bilinear generating function for Askey—Wilson polynomials involving the Poisson kernel of Al-Salam and Chihara polynomials. July 6, 1997. Date accepted: September 23, 1998.     

On formulas of Ramanujan and Evans

We give short elementary proofs of a formula of Ramanujan as interpreted by Bradley, and a companion formula originally proved by W. Chu. We also give an elementary proof of a generalization of an identity originally proved using modular functions and used to study a generating function for the number of partitions with specified crank. Research partially supported by NSF grant DMS 99-70865. 2000 Mathematics Subject Classification Primary—39A70, 11P83; Secondary—33C20     

Estimating a density on the positive half line by the method of orthogonal series

The kernel method of density estimation is not so attractive when the density has its support confined to (0, ∞), particularly when the density is unsmooth at the origin. In this situation the method of orthogonal series is competitive. We consider three essentially different orthogonal series—those based on the even and odd Hermite functions, respectively, and that based on Laguerre functions—and compare them from the point of view of mean integrated square error.     

On the number of primitive representations of integers as sums of squares

Formulas for the number of primitive representations of any integer n as a sum of k squares are given, for 2 ≤ k ≤ 8, and for certain values of n, for 9 ≤ k ≤ 12. The formulas have a similar structure and are striking for their simplicity. Dedicated to Richard Askey on the occasion of his 70th birthday. 2000 Mathematics Subject Classification Primary—11E25; Secondary—05A15, 33E05.     

Existence and Asymptotic Behavior of Radially Symmetric Solutions to a Semilinear Hyperbolic System in Odd Space Dimensions

This paper is concerned with a class of semilinear hyperbolic systems in odd space dimensions. Our main aim is to prove the existence of a small amplitude solution which is asymptotic to the free solution as t→-∞in the energy norm, and to show it has a free profile as t→ ∞. Our approach is based on the work of [11]. Namely we use a weighted L∞norm to get suitable a priori estimates. This can be done by restricting our attention to radially symmetric solutions. Corresponding initial value problem is also considered in an analogous framework. Besides, we give an extended result of [14] for three space dimensional case in Section 5, which is prepared independently of the other parts of the paper.     

Bar weights of bar partitions and spin character zeros

The main combinatorial result in this article is a classification of bar partitions of n which are of maximal p-bar weight for all odd primes p ≤ n. As a consequence, we show that apart from very few exceptions any irreducible spin character of the double covers of the symmetric and alternating groups vanishes on some element of odd prime order. Mathematics Subject Classification 05A17, 20C30 An extended abstract for this paper appeared in the Proceedings of the FPSAC’06 conference.     

New transformations for elliptic hypergeometric series on the root system A n

Recently, Kajihara gave a Bailey-type transformation relating basic hypergeometric series on the root system A _n , with different dimensions n. We give, with a new, elementary proof, an elliptic extension of this transformation. We also obtain further Bailey-type transformations as consequences of our result, some of which are new also in the case of basic and classical hypergeometric series. 2000 Mathematics Subject Classification Primary—33D67; Secondary—11F50     

On identities of the Rogers-Ramanujan type

A generalized Bailey pair, which contains several special cases considered by Bailey (Proc. London Math. Soc. (2), 50, 421–435 (1949)), is derived and used to find a number of new Rogers-Ramanujan type identities. Consideration of associated q-difference equations points to a connection with a mild extension of Gordon’s combinatorial generalization of the Rogers-Ramanujan identities (Amer. J. Math., 83, 393–399 (1961)). This, in turn, allows the formulation of natural combinatorial interpretations of many of the identities in Slater’s list (Proc. London Math. Soc. (2) 54, 147–167 (1952)), as well as the new identities presented here. A list of 26 new double sum–product Rogers-Ramanujan type identities are included as an Appendix. 2000 Mathematics Subject Classification Primary—11B65; Secondary—11P81, 05A19, 39A13