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1.
We study several statistics for integer partitions: for a random partition of an integer n we consider the average size of the smallest gap (missing part size), the multiplicity of the largest part, and the largest
repeated part size. Furthermore, we estimate the number of gap-free partitions of n.
2000 Mathematics Subject Classification Primary—05A17; Secondary—11P82
Dedicated to Helmut Prodinger on the occasion of his 50th birthday
P.J. Grabner is supported by the START-project Y96-MAT of the Austrian Science Fund.
This material is based upon work supported by the National Research Foundation under grant number 2053740. 相似文献
2.
In previous work arising from the study of Ramanujan's Lost Notebook, a new Abel type lemma was proved. In this paper, we
discuss extensions of this lemma and use it to prove many q-series identities.
The first author was partially supported by NSF grant DMS–0200047. The second author was partially supported by FCT, Portugal,
through program POCTI.
2000 Mathematics Subject Classification:Primary—33D15; Secondary—05A30 相似文献
3.
We deduce several curious q-series expansions by applying inverse relations to certain identities for basic hypergeometric series. After rewriting some
of these expansions in terms of q-integrals, we obtain, in the limit q→ 1, some curious beta-type integral evaluations which appear to be new.
Dedicated to Dick Askey on the occasion of his 70th birthday.
2000 Mathematics Subject Classification Primary—15A09, 33D15, 33E20; Secondary—05A30
M. Schlosser was fully supported by an APART fellowship of the Austrian Academy of Sciences. 相似文献
4.
By jagged partitions we refer to an ordered collection of non-negative integers (n1, n2,..., nm) with nm≥ p for some positive integer p, further subject to some weakly decreasing conditions that prevent them for being genuine partitions. The case analyzed in
greater detail here corresponds to p = 1 and the following conditions ni≥ ni+1−1 and ni≥ ni+2. A number of properties for the corresponding partition function are derived, including rather remarkable congruence relations.
An interesting application of jagged partitions concerns the derivation of generating functions for enumerating partitions
with special restrictions, a point that is illustrated with various examples.
2000 Mathematics Subject Classification: Primary—05A15, 05A17, 05A19 相似文献
5.
Kazuhiro Hikami 《The Ramanujan Journal》2006,11(2):175-197
Studied is a generalization of Zagier’s q-series identity. We introduce a generating function of L-functions at non-positive integers, which is regarded as a half-differential of the Andrews-Gordon q-series. When q is a root of unity, the generating function coincides with the quantum invariant for the torus knot.
2000 Mathematics Subject Classification Primary—11F67, 57M27, 05A30, 11F23 相似文献
6.
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. 相似文献
7.
We consider rational functions with n prescribed poles for which there exists a divided difference operator transforming them to rational functions with n−1 poles. The poles of such functions are shown to lie on the elliptic grids. There is a one-to-one correspondence between
this problem of admissible grids and the Poncelet problem on two quadrics. Additionally, we outline an explicit scheme of
the Padé interpolation with prescribed poles and zeros on the elliptic grids.
Dedicated to Richard Askey on the occasion of his seventieth birthday.
2000 Mathematics Subject Classification Primary—42C05; Secondary—39A13, 41A05, 41A21. 相似文献
8.
Cilanne E. Boulet 《The Ramanujan Journal》2006,12(3):315-320
We present a new partition identity and give a combinatorial proof of our result. This generalizes a result of Andrews in
which he considers the generating function for partitions with respect to size, number of odd parts, and number of odd parts
of the conjugate.
2000 Mathematics Subject Classification Primary—05A17; Secondary—11P81 相似文献
9.
Jeremy Rouse 《The Ramanujan Journal》2006,12(2):197-205
We provide uniform formulas for the real period and the trace of Frobenius associated to an elliptic curve in Legendre normal
form. These are expressed in terms of classical and Gaussian hypergeometric functions, respectively.
2000 Mathematics Subject Classification Primary—11G05, 33C05
This research was supported by K. Ono’s NSF grant 相似文献
10.
Stamatis Koumandos 《The Ramanujan Journal》2007,14(1):1-38
We establish a best possible extension of a famous Theorem of Vietoris about the positivity of a general class of cosine sums.
Our result refines and sharpens several earlier generalizations of this Theorem, and settles some open questions regarding
the possibility of further improvement of it. Some new inequalities for trigonometric sums are given. We show that our results
have applications within the context of positive sums of Gegenbauer polynomials and quadrature methods. We also obtain some
existing estimates for the location of zeros of certain trigonometric polynomials under a weakened condition on their coefficients.
2000 Mathematics Subject ClassificationPrimary—42A05; Secondary—42A32,26D05, 26D15, 33B15, 33C45
Dedicated to Richard Askey on the occasion of his 70th birthday 相似文献
11.
Andrew V. Sills 《The Ramanujan Journal》2006,11(3):403-429
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 相似文献
12.
Explicit expressions for restricted partition function W(s,dm) and its quasiperiodic components Wj(s,dm) (called Sylvester waves) for a set of positive integers dm = {d1, d2, ..., dm} are derived. The formulas are represented in a form of a finite sum over Bernoulli and Eulerian polynomials of higher order
with periodic coefficients. A novel recursive relation for the Sylvester waves is established. Application to counting algebraically
independent homogeneous polynomial invariants of finite groups is discussed.
2000 Mathematics Subject Classification Primary—11P81; Secondary—11B68, 11B37
The research was supported in part (LGF) by the Gileadi Fellowship program of the Ministry of Absorption of the State of Israel. 相似文献
13.
Warren P. Johnson 《The Ramanujan Journal》2007,13(1-3):167-201
More than 200 years ago, Pfaff found two generalizations of Leibniz’s rule for the nth derivative of a product of two functions. Thirty years later Cauchy found two similar identities, one equivalent to one
of Pfaff’s and the other new. We give simple proofs of these little-known identities and some further history. We also give
applications to Abel-Rothe type binomial identities, Lagrange’s series, and Laguerre and Jacobi polynomials. Most importantly,
we give extensions that are related to the Pfaff/Cauchy theorems as Hurwitz’s generalized binomial theorems are to the Abel-Rothe
identities. We apply these extensions to Laguerre and Jacobi polynomials as well.
Dedicated to Dick Askey on the occasion of his 70th birthday.
2000 Mathematics Subject Classification Primary—05A19; Secondary—33C45 相似文献
14.
Wenchang Chu 《The Ramanujan Journal》2007,13(1-3):221-225
The Pfaff-Euler Transform for hypergeometric 2
F
1-series is applied to provide a direct and elementary proof that the hypergeometric representation with algebraic parameters
of Pollaczek polynomials are indeed polynomials.
Dedicated to Richard Askey on the occasion of his 70th birthday.
2000 Mathematics Subject Classification Primary—33C45; Secondary—33C05 相似文献
15.
Using the theory of Macdonald polynomials, a number of q-integrals of Selberg type are proved.
2000 Mathematics Subject Classification: Primary—33D05, 33D52, 33D60 相似文献
16.
We apply method of separation of the variables to q-analogs of several equations of mathematical physics.
Dedicated to Dick Askey on his 70th birthday.
2000 Mathematics Subject Classification Primary—33D45, 42C10; Secondary—33D15. 相似文献
17.
Christine Bessenrodt 《Journal of Algebraic Combinatorics》2007,26(1):107-124
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. 相似文献
18.
In 1991 Tratnik derived two systems of multivariable orthogonal Racah polynomials and considered their limit cases. q-Extensions of these systems are derived, yielding systems of multivariable orthogonal q-Racah polynomials, from which systems of multivariable orthogonal q-Hahn, dual q-Hahn, q-Krawtchouk, q-Meixner, and q-Charlier polynomials follow as special or limit cases.
Dedicated to Richard Askey on the occasion of his 70th birthday.
2000 Mathematics Subject Classification Primary—33D50; Secondary—33C50
Supported in part by NSERC grant #A6197. 相似文献
19.
Kevin W. J. Kadell 《The Ramanujan Journal》2007,13(1-3):449-469
We use telescoping partial fractions decompositions to give new proofs of the orthogonality property and the normalization
relation for the little q-Jacobi polynomials, and the q-Saalschütz sum. In [20], we followed the development [19] of Schur functions for partitions with complex parts, and we showed
that there exist natural little q-Jacobi functions of complex order which satisfy extensions of the orthogonality property and normalization relation of the
little q-Jacobi polynomials, and that these two results follow from and together imply the nonterminating form of the q-Saalschütz sum. Writing the q-Pochhammer symbol of complex order as a ratio of infinite products in the usual way, we obtain new telescoping partial fractions
decomposition proofs of our results [20] for the little q-Jacobi functions of complex order. We give several new proofs of the q-Saalschütz sum and its nonterminating form.
For our friends Dick and Liz
2000 Mathematics Subject Classification Primary—42C05; Secondary—33C45, 33C47 相似文献
20.
Hjalmar Rosengren 《The Ramanujan Journal》2006,12(2):155-166
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 相似文献