A Generalization of Partition Identities for First Differences of Partitions of $n$ Into at Most $m$ Parts

We show for a prime power number of parts $m$ that the first differences of partitions into at most $m$ parts can be expressed as a non-negative linear combination of partitions into at most $m-1$ parts. To show this relationship, we combine a quasipolynomial construction of $p(n,m)$ with a new partition identity for a finite number of parts. We prove these results by providing combinatorial interpretations of the quasipolynomial of $p(n,m)$ and the new partition identity. We extend these results by establishing conditions for when partitions of $n$ with parts coming from a finite set $A$ can be expressed as a non-negative linear combination of partitions with parts coming from a finite set $B$.

Integral and series representations of q-Polynomials and functions: Part III theta, Ramanujan and q-Bessel functions

In this paper, we use an identity connecting a modified [Formula: see text]-Bessel function and a [Formula: see text] function to give [Formula: see text]-versions of entries in the Lost Notebook of Ramanujan. We also establish an identity which gives an [Formula: see text]-version of a partition identity. We prove new relations and identities involving theta functions, the Ramanujan function, the Stieltjes–Wigert, [Formula: see text]-Lommel and [Formula: see text]-Bessel polynomials. We introduce and study [Formula: see text]-analogues of the spherical Bessel functions.

Bounds for d-distinct partitions

International audience Euler's identity and the Rogers-Ramanujan identities are perhaps the most famous results in the theory of partitions. According to them, 1-distinct and 2-distinct partitions of n are equinumerous with partitions of n into parts congruent to ±1 modulo 4 and partitions of n into parts congruent to ±1 modulo 5, respectively. Furthermore, their generating functions are modular functions up to multiplication by rational powers of q. For d ≥ 3, however, there is neither the same type of partition identity nor modularity for d-distinct partitions. Instead, there are partition inequalities and mock modularity related with d-distinct partitions. For example, the Alder-Andrews Theorem states that the number of d-distinct partitions of n is greater than or equal to the number of partitions of n into parts which are congruent to ±1 (mod d+3). In this note, we present the recent developments of generalizations and analogs of the Alder-Andrews Theorem and establish asymptotic lower and upper bounds for the d-distinct partitions. Using the asymptotic relations and data obtained from computation, we propose a conjecture on a partition inequality that gives an upper bound for d-distinct partitions. Specifically, for d ≥ 4, the number of d-distinct partitions of n is less than or equal to the number of partitions of n into parts congruent to ±1 (mod m), where m ≤ 2dπ^2 / [3 log^2 (d)+6 log d] .

Proofs of some partition identities conjectured by Kanade and Russell

Kanade and Russell conjectured several Rogers–Ramanujan-type partition identities, some of which are related to level 2 characters of the affine Lie algebra $$A_9^{(2)}$$ A 9 ( 2 ) . Many of these conjectures have been proved by Bringmann, Jennings-Shaffer and Mahlburg. We give new proofs of five conjectures first proved by those authors, as well as four others that have been open until now. Our proofs for the new cases use quadratic transformations for Askey–Wilson and Rogers polynomials. We also obtain some related results, including a partition identity conjectured by Capparelli and first proved by Andrews.

In Search of Lost Glory

This book traces the trajectory of Sindhi nationalism in its quest for lost glory. It examines the Sindhi nationalist movement through its various stages, ranging from pre-partition identity construction in pursuit of the separation of Sindh from Bombay, to the post-partition travails of a community which lost its identity and its capital as a result of the arrival of millions of migrants from India (Muhajirs) and of the actions of an over-bearing central government. Going beyond the state and its power play, the book examines the long history of Sindhi-Muhajir contestation for resources in the post-partition period. The book develops a comprehensive profile of the agency of nationalist parties in Sindh, including the Sindhudesh detour and the later fragmentation of the Jiye Sind movement, which was followed by the emergence of new parties. The author also analyzes the dual role of the Pakistan Peoples Party (PPP) as an ethnic entrepreneur inside the province while operating as a federal party outside Sindh. The book covers nationalist contention at three levels: the struggle for power between Sindh and a dominant Centre; the inter-ethnic conflict between Sindhis and Muhajirs; and the intra-ethnic contestation between the Sindhi nationalists themselves and the PPP.

Partitions associated with two fifth-order mock theta functions and Beck-type identities

In this paper, we first generalize Ramanujan’s partition identity derived from the fifth order mock theta functions [Formula: see text] and [Formula: see text]. Then we establish Beck-type identities based on our general partition identity. All the results are proved both analytically and combinatorially.

Almost partition identities

An almost partition identity is an identity for partition numbers that is true asymptotically100%of the time and fails infinitely often. We prove a kind of almost partition identity, namely that the number of parts in all self-conjugate partitions of n is almost always equal to the number of partitions of n in which no odd part is repeated and there is exactly one even part (possibly repeated). Not only does the identity fail infinitely often, but also, the error grows without bound. In addition, we prove several identities involving the number of parts in restricted partitions. We show that the difference in the number of parts in all self-conjugate partitions of n and the number of parts in all partitions of n into distinct odd parts equals the number of partitions of n in which no odd part is repeated, the smallest part is odd, and there is exactly one even part (possibly repeated). We provide both analytic and combinatorial proofs of this identity.