共查询到20条相似文献,搜索用时 0 毫秒
1.
Michael D. Hirschhorn 《The Ramanujan Journal》2013,31(1-2):15-22
B. Yuttanan has shown that certain identities of Ramanujan and of Horie and Kanou can be factorized, that is, obtained by multiplying together certain pairs of identities. We give alternative, simpler factorizations of the same identities, and more. 相似文献
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Pee Choon Toh 《Discrete Mathematics》2012,312(6):1244-1250
Using elementary methods, we establish several new Ramanujan type identities and congruences for certain pairs of partition functions. 相似文献
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The Ramanujan Journal - In this paper, we represent the generating function of the rank function as a summation of four parts—a constant, two Lambert series and a product. Applying it to... 相似文献
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Krishnaswami Alladi George E. Andrews Alexander Berkovich 《Inventiones Mathematicae》2003,153(2):231-260
We prove a new four parameter q-hypergeometric series identity from which the three parameter identity for the Göllnitz theorem due to Alladi, Andrews, and Gordon follows as a special case by setting one of the parameters equal to 0. The new identity is equivalent to a four parameter partition theorem which extends the deep theorem of Göllnitz and thereby settles a problem raised by Andrews thirty years ago. Some consequences including a quadruple product extension of Jacobis triple product identity, and prospects of future research are briefly discussed. Mathematics Subject Classification (1991) 05A15, 05A17, 05A19, 11B65, 33D15 相似文献
5.
K. Vishnu Namboothiri 《The Ramanujan Journal》2017,44(3):531-547
We derive certain identities involving various known arithmetical functions and a generalized version of Ramanujan sum. L. Tóth constructed certain weighted averages of Ramanujan sums with various arithmetic functions as weights. We choose a generalization of Ramanujan sum given by E. Cohen and derive the weighted averages corresponding to the versions of the weighted averages established by Tóth. 相似文献
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The terminating basic hypergeometric series is investigated through the modified Abel lemma on summation by parts. Numerous known summation and transformation formulae are derived in a unified manner. Several new identities for the terminating quadratic, cubic and quartic series are also established. 相似文献
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The main purpose of this paper is to introduce new sums that are analogous to Dedekind sums. Using analysis and properties of Dirichlet \(L\) -functions, we study mean values for these new sums, and give a sharper mean value formula for it. 相似文献
10.
Heng Huat Chan 《Journal of Number Theory》2010,130(9):1898-1913
We establish several new analogues of Ramanujan's exact partition identities using the theory of modular functions. 相似文献
11.
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
12.
Let p = p(a, b, c) be the number of partitions of a into b parts, no part exceeding c. Bellavitis and perhaps some earlier writers noted that p satisfies three very simple identities. Here p is generalized to a function of k + 1 variables in a natural way. One of the identities then generalizes; the proof of this (which depends on the P. Hall commutator collecting process) is given only for k = 3. 相似文献
13.
Krishnaswami Alladi 《The Ramanujan Journal》2013,31(1-2):213-238
Utilizing a six-variable extension of Heine’s q-hypergeometric transformation that we previously obtained, we now derive variants of Heine’s transformation formula and the Lebesgue identity. The variant of Cauchy’s identity also obtained by us earlier is crucial in these derivations. We then establish some new partition identities which are variants of, and shed new light on, some fundamental classical partition identities. 相似文献
14.
Jeffrey B Remmel 《Journal of Combinatorial Theory, Series A》1982,33(3):273-286
A bijective proof of a general partition theorem is given which has as direct corollaries many classical partition theorems due to Euler, Glaisher, Schur, Andrews, Subbarao, and others. It is shown that the bijective proof specializes to give bijective proofs of these classical results and moreover the bijections which result often coincide with bijections which have occurred in the literature. Also given are some sufficient conditions for when two classes of words omitting certain sequences of words are in bijection. 相似文献
15.
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
16.
Krishnaswami Alladi 《The Ramanujan Journal》2010,23(1-3):227-241
We provide a simple proof of a partial theta identity of Andrews and study the underlying combinatorics. This yields a weighted partition theorem involving partitions into distinct parts with smallest part odd which turns out to be a companion to a weighted partition theorem involving the same partitions that we recently deduced from a partial theta identity in Ramanujan’s Lost Notebook. We also establish some new partition identities from certain special cases of Andrews’ partial theta identity. 相似文献
17.
Lisa Lorentzen 《The Ramanujan Journal》2008,17(3):369-385
We present an idea on how Ramanujan found some of his beautiful continued fraction identities. Or more to the point: why he
chose the ones he wrote down among all possible identities.
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Let p(n) denote the number of partitions of n. Recall Ramanujan’s three congruences for the partition function,
These congruences have been proven via q-series identities, combinatorial arguments, and the theory of Hecke operators. We present a new proof which does not rely
on any specialized identities or combinatorial constructions, nor does it necessitate introducing Hecke operators. Instead,
our proof follows from simple congruences between the coefficients of modular forms, basic properties of Klein’s modular j-function, and the chain rule for differentiation. Furthermore, this proof naturally encompasses all three congruences in
a single argument.
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