Krishnaswami Alladi

Partition identities involving gaps and weights

We obtain interesting new identities connecting the famous partition functions of Euler, Gauss, Lebesgue, Rogers–Ramanujan and others by attaching weights to the gaps between parts. The weights are in general multiplicative. Some identities involve powers of 2 as weights and yield combinatorial information about some remarkable partition congruences modulo powers of 2.

Generalizations of Schur's partition theorem

AbstractSchurs partition theorem states that the number of partitions ofn into distinct parts ≡ 1,2 (mod 3) is equal to the number of partitions ofn into parts with minimal difference 3 and no consecutive multiples of 3. A three-parameter generalization of Gleissbergs refinement of Schurs theorem is obtained by showing that

Partition identities and a continued fraction of Ramanujan

\Pi _{m = 1}^\infty (1 + aq^m )(1 + bq^m )

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

is equal to the numerator of a certain continued fraction. Two proofs are presented, one completely combinatorial, and one using generating functions.

A combinatorial correspondence related to Göllnitz’ (big) partition theorem and applications

- Preface Part I. The Legacy of Alladi Ramakrishnan - Contributions of Alladi Ramakrishnan to the Mathematical Sciences. - Alladi Ramakrishnans Theoretical Physics Seminar. - Telegrams received for the MATSCIENCE inauguration. - The miracle has happened, speech given by Alladi Ramakrishnan at the inauguration of MATSCIENCE. - Overseas Trips of Alladi Ramakrishnan. - List of publications of Alladi Ramakrishnan. - List of PhD students of Alladi Ramakrishnan. Part II. Pure Mathematics - Inversion and invariance of characteristic terms - part I (Abhyankar). - Partitions with non-repeating odd parts and q-hypergeometric identities (Alladi). - q-Catalan identities (Andrews). - Completing Brahmaguptas extension of Ptolemys theorem (Askey). - A transformation formula involving the gamma and Riemann zeta functions in Ramanujans Lost Notebook (Berndt and Dixit). - Ternary quadratic forms, modular equations, and certain positivity conjectures (Berkovich and Jagy). - How often is n! a sum of three squares? (Deshouillers and Luca).- Crystal symmetry viewed as zeta symmetry - II (Kanemitsu and Tsukuda). - Eulerian polynomials: From Eulers time to the present (Foata). - Crystal symmetry viewed as zeta symmetry II (Kanemitsu and Tsukada). - Positive homogeneous minima for a system of linear forms (Raghavan). - The divisor matrix, Dirichlet series, and SL(2,Z)(Sin and Thompson). - Proof of a conjecture of Alladi Ramakrishnan on circulants (Waldschmidt). Part III. Probability and Statistics - Branching random walks (Athreya). - A commentary on the logistic distribution (Ghosh, Choi, and Li). - Entropy and cross entropy characterizations and applications (Rao). - Optimal weights for a class of rank tests for censored bivariate data (Rao, Raychaudhuri, and Wu). - Connections between Bernoulli strings and random permutations (Sethuraman and Sethuraman). - Storage models for a class of master equations with separable kernels (Vittal, Jayasankar, and Muralidhar). Part IV. Theoretical Physics and Applied Mathematics - Inverse consistent deformable image registration (Chen and Ye). - A statistical model for the quark structure of the nucleon (Devanathan and Karthiyayini). - On generalized Clifford algebras and their physical applications (Jagannathan). - (p,q)-Rogers-Szego polynomial and the (p,q)-oscillator (Jagannathan and Sridhar). - Rethinking renormalization (Klauder). - Magnetism, FeS celluloids, and origins of life (Mitra-Delmotte and Mitra). - The Ehrenfest theorem in quantum field theory (Parthasarathy).

A Fundamental Invariant in the Theory of Partitions

Abstract We study the numerator and denominator of a continued fraction R(a, b) of Ramanujan and establish the equality of various restricted partition functions. We use the continued fraction to give a unified approach to several partition identities some of which generalize results of Bressoud and Gollnitz. We also give a combinatorial interpretation for the coefficients in the power series expansion of the reciprocal 1 R (−a, −b) , extending a result of Odlyzko and Wilf.

Schur’s partition theorem, companions, refinements and generalizations

-Preface (K. Alladi and F. Garvan).- 1. MacMahons dream (G. E. Andrews and P. Paule).- 2. Ramanujans elementary method in partition congruences (B. Berndt, C. Gugg, and S. Kim).- 3. Coefficients of harmonic Maass forms (K. Bringmann and K. Ono).- 4. On the growth of restricted partition functions (E. R. Canfield and H. Wilf).- 5. On applications of roots of unity to product identities (Z. Cao).- 6. Lecture hall sequences, q-series, and asymmetric partition identities (S. Corteel, C. Savage and A. Sills).- 7. Generalizations of Hutchinsons curve and the Thomae formula (H. Farkas).- 8. On the parity of the Rogers-Ramanujan coefficients (B. Gordon).- 9. A survey of the classical mock theta functions (B. Gordon and R. McIntosh).- 10. An application of the Cauchy-Sylvester theorem on compound determinants to a BC_n Jackson integral (M. Ito and S. Okada).- 11. Multiple generalizations of q-series identities found in Ramanujans Lost Notebook (Y. Kajihara).- 12. Non-terminating q-Whipple transformations for basic hypergeometric series in U(n) (S. C. Milne and J. W. Newcomb).

Multiplicative Functions and Small Divisors, II

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 Jacobi’s triple product identity, and prospects of future research are briefly discussed.