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Krishnaswami Alladi

Publications and source records attributed to Krishnaswami Alladi.

14 recordsLinked to original sources
arXiv2026

Capparelli's partition theorem as part of an infinite hierarchy: Combinatorial and Weighted Words extensions of recent work

In a recent paper, the authors introduced an infinite hierarchy of $q$-hypergeometric identities, of which the first three orders, $0$, $1$, and $2$, relate to the partition theorems of Euler, Lebesgue, and Capparelli, and stated a partition theorem at order 4 which lies beyond Capparelli's theorem. Here, we first state certain partition theorems that hold at all even orders beyond Capparelli and provide bijective proofs for these theorems. In doing so, we show that there is a fourfold infinite hierarchy of partition theorems that emanates from Capparelli's theorem, which is the base case. It is also shown that the equality of two of the four generating functions holds for all orders, odd and even. Lastly, a very general framework for the remaining two functions is constructed via the method of weighted words, encompassing all possible orders and yielding several infinite hierarchies with different dilations and translations.

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arXiv2026

Duality Between Prime Factors and The Prime Number Theorem For Arithmetic Progressions -- Higher Order Dualities

In 1977, the first author observed a duality between the largest and smallest prime factors of integers, and established as a consequence some new results on the M\"obius function $\mu(n)$ using the Prime Number Theorem for Arithmetic Progressions. In that 1977 paper, higher order dualities were observed involving the $k$-th largest and $k$-th smallest prime factors, facilitated by the M\"obius function and $\omega(n)^{k-1}$, where $\omega(n)$ is the number of distinct prime factors on $n$. In 2024, the first author and Jason Johnson proved new results involving $\mu(n)$ and $\omega(n)$, by exploiting the second order duality identity of Alladi (1977). We establish here extensions to all higher orders $k$, the results of Alladi (1977) and of Alladi-Johnson (2024), by utilizing the $k$-th order duality in Alladi's 1977 paper. First, we show that for each $k\geq 2$, $$ \sum_{n=2}^{\infty} \frac{\mu(n)\omega(n)^{k}}{n} =0, $$ where $\mu(n)$ is the M\"obius Function and $\omega(n)$ counts the number of distinct prime factors of $n$. Further, using the General Duality Identity and the Prime Number Theorem of Arithmetic Progressions, we prove that for integers $j,\ell$ satisfying $1 \leq j \leq \ell$ and $(j,\ell)=1$ $$ \sum_{\substack{n=2 \\ p_1(n) \equiv j\;(mod\;\ell)}}^{\infty} \frac{\mu(n)\omega(n)^{k-1}}{n}=0, \nonumber $$ for every $k \geq 3$; this result for $k=1$ is due to Alladi (1977) and for $k=2$ due to Alladi-Johnson (2024). We also recast this result in the following manner as a density-type theorem: for integers $j,\ell$ satisfying $1 \leq j \leq \ell$ and $(j,\ell)=1$ $$ (-1)^k\sum_{\substack{n=2 \\ p_1(n) \equiv j\;(mod\;\ell)}}^{\infty} \frac{\mu(n){\omega(n)-1 \choose k-1}}{n}=\frac{1}{\varphi(\ell)}, \nonumber $$ for every $k \geq 3$. All results are established here in quantitative form.

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arXiv2025

Basis partitions and their signature

Basis partitions are minimal partitions corresponding to successive rank vectors. We show combinatorially how basis partitions can be generated from primary partitions which are equivalent to the Rogers-Ramanujan partitions. This leads to the definition of a signature of a basis partition that we use to explain certain parity results. We then study a special class of basis partitions which we term as complete. Finally we discuss basis partitions and minimal basis partitions among partitions with non-repeating odd parts by representing them using 2-modular graphs.

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arXiv2025

Some $q$-hypergeometric identities associated with partition theorems of Lebesgue, Schur and Capparelli

Here, we establish a polynomial identity in three variables $a, b, c$, and with the degree of the polynomial given in terms of two integers $L, M$. By letting $L$ and $M$ tend to infinity, we get the 1993 Alladi-Gordon $q$-hypergeometric key-identity for the generalized Schur Theorem as well as the fundamental Lebesgue identity by two different choices of the variables. This polynomial identity provides a generalization and a unified approach to the Schur and Lebesgue theorems. We discuss other analytic identities for the Lebesgue and Schur theorems and also provide a key identity ($q$-hypergeometric) for Andrews' deep refinement of the Alladi-Schur theorem. Finally, we discuss a new infinite hierarchy of identities, the first three of which relate to the partition theorems of Euler, Lebesgue, and Capparelli, and provide their polynomial versions as well.

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arXiv2024

Parity results concerning the generalized divisor function involving small prime factors of integers

Let $\nu_y(n)$ denote the number of distinct prime factors of $n$ that are $<y$. For $k$ a positive integer, and for $k+2\leq y\leq x$, let $S_{-k}(x,y)$ denote the sum \begin{eqnarray*} S_{-k}(x,y):=\sum_{n\leq x}(-k)^{\nu_y(n)}. \end{eqnarray*} In this paper, we describe our recent results on the asymptotic behavior of $S_{-k}(x,y)$ for $k+2\leq y\leq x$, and $x$ sufficiently large. There is a crucial difference in the asymptotic behavior of $S_{-k}(x,y)$ when $k+1$ is a prime and $k+1$ is composite, and this makes the problem particularly interesting. The results are derived utilizing a combination of the Buchstab-de Bruijn recurrence, the Perron contour integral method, and certain difference-differential equations. We present a summary of our results against the background of earlier work of the first author on sums of the M\"{o}bius function over integers with restricted prime factors and on a multiplicative generalization of the sieve.

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arXiv2024

Duality between prime factors and the Prime Number Theorem for Arithmetic Progressions -- II

In the first paper under this title (1977), the first author utilized a duality identity between the largest and smallest prime factors involving the Moebius function, to establish the following result as a consequence of the Prime Number Theorem for Arithmetic Progressions: If $k$ and $\ell$ are positive integers, with $1\le\ell\le k$ and $(\ell, k)=1$, then $$ \sum_{n\ge 2,\, p(n)\equiv\ell(mod\,k)}\frac{\mu(n)}{n}=\frac{-1}{\phi(k)}, $$ where $\mu(n)$ is the Moebius function, $p(n)$ is the smallest prime factor of $n$, and $\phi(k)$ is the Euler function. Here we utilize the next level Duality identity between the second largest prime factor and the smallest prime factor, involving the Moebius function and $\omega(n)$, the number of distinct prime factors of $n$, to establish the following result as a consequence of the Prime Number Theorem for Arithmetic Progressions: For all $\ell$ and $k$ as above, $$ \sum_{n\ge 2, \, p(n)\equiv\ell(mod\,k)}\frac{\mu(n)\omega(n)}{n}=0. $$ A quantitative version of this result is proved.

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arXiv2004

Goellnitz-Gordon partitions with weights and parity conditions

A Goellnitz-Gordon partition is one in which the parts differ by at least 2, and where the inequality is strict if a part is even. Let Q_i(n) denote the number of partitions of n into distinct parts not congruent to i mod 4. By attaching weights which are powers of 2 and imposing certain parity conditions on Goellnitz-Gordon partitions, we show that these are equinumerous with Q_i(n) for i=0,2. These complement results of Goellnitz on Q_i(n) for i=1,3, and of Alladi who provided a uniform treatment of all four Q_i(n), i=0,1,2,3, in terms of weighted partitions into parts differing by >= 4. Our approach here provides a uniform treatment of all four Q_i(n) in terms of certain double series representations. These double series identities are part of a new infinite hierarchy of multiple series identities.

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arXiv2003

A new four parameter q-series identity and its partition implications

We prove a new four parameter q-hypergeometric series identity from which the three parameter key identity for the Goellnitz 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 Goellnitz and thereby settles a problem raised by Andrews thirty years ago. Some consequences including a quadruple product extension of Jacobi's triple product identity, and prospects of future research are briefly discussed.

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arXiv2002

A limiting form of the q-Dixon_4ϕ_3 summation and related partition identities

By considering a limiting form of the q-Dixon_4ϕ_3 summation, we prove a weighted partition theorem involving odd parts differing by >= 4. A two parameter refinement of this theorem is then deduced from a quartic reformulation of Goellnitz's (Big) theorem due to Alladi, and this leads to a two parameter extension of Jacobi's triple product identity for theta functions. Finally, refinements of certain modular identities of Alladi connected to the Goellnitz-Gordon series are shown to follow from a limiting form of the q-Dixon_4ϕ_3 summation.

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arXiv2002

New polynomial analogues of Jacobi's triple product and Lebesgue's identities

In a recent paper by the authors, a bounded version of Goellnitz's (big) partition theorem was established. Here we show among other things how this theorem leads to nontrivial new polynomial analogues of certain fundamental identities of Jacobi and Lebesgue. We also derive a two parameter extension of Jacobi's famous triple product identity.

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arXiv2000

New Weighted Rogers-Ramanujan Partition Theorems and their Implications

This paper has a two-fold purpose. First, by considering a reformulation of a deep theorem of Göllnitz, we obtain a new weighted partition identity involving the Rogers-Ramanujan partitions, namely, partitions into parts differing by at least two. Consequences of this include Jacobi's celebrated triple product identity for theta functions, Sylvester's famous refinement of Euler's theorem, as well as certain weighted partition identities. Next, by studying partitions with prescribed bounds on successive ranks and replacing these with weighted Rogers-Ramanujan partitions, we obtain two new sets of theorems - a set of three theorems involving partitions into parts $\not\equiv 0, \pm i$ ($mod$ 6), and a set of three theorems involving partitions into parts $\not\equiv 0, \pm i$ ($mod$ 7), $i=1,2,3$.

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arXiv2000

A four parameter generalization of Gollnitz's (BIG) partition theorem

We announce a new four parameter partition theorem from which the (big) theorem of Gollnitz follows by setting any one of the parameters equal to 0. This settles a problem of Andrews who asked whether there exists a result that goes beyond the partition theorem of Gollnitz. We state a four parameter q-series identity (key identity) which is the generating function form of this theorem. In a subsequent paper, the proof of the new four parameter key identity will be given.

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