Research

Research Interests

(under constant development)

My research interest are divided into three main areas:

1. Representations of affine Lie algebras, vertex theoretic structures and connections with integer partition theory;

1a. Mathematics related to Capparelli’s identities on partitions;

2. Combinatorics of orthogonal polynomials and generalizations, integer sequences;

3. Span of Hyperbolic Polynomials: algebraic integers and related topics.

Some of my papers classified according to the above three areas:

Topic 1: a. Capparelli, Lepowsky, Milas, The Rogers-Selberg recursions, the Gordon-Andrews identities and intertwining operators, Ramanujan J. 12 (2006), no. 3, 379-397.

b. Capparelli, On some theorems of Hirschhorn, Comm. Algebra 32 (2004), no 2, 629-635

c. Capparelli, Lepowsky, Milas, The Rogers-Ramanujan recursion and intertwining operators, Commun. Contemp. Math. 5 (2003), no. 6, 947-966

d. Capparelli, A construction of the level 3 modules for the affine Lie algebra A₂⁽²⁾ and a new combinatorial identity of the Rogers-Ramanujan type. Trans. Amer. Math. Soc. 348 (1996), no. 2, 481-501

e. Capparelli, On some representations of twisted affine Lie algebras and combinatorial identities, J. Algebra 154 (1993), no. 2, 335-355

Topic 2: a. Capparelli, Root power sums and Chebyshev polynomials, Rocky Mountain J. Math, 48 (2018), no. 1, 59-74

b. Capparelli, Ferrari, Munarini, Zagaglia, A generalization of the “problem des rencontres”. J. Integer Seq. 21 (2018), no 2.

c. Capparelli, Del Fra, Dyck paths, Motzkin paths, aand the binomial transform, J. Integer Seq. 18 (2015), no. 8

d. Capparelli, Maroscia, On two sequences of orthogonal polynomials related to Jordan blocks, Mediterr. J. Math. 10 (2013), no. 4, 1609-1630.

e. Capparelli, Del Fra, Pepe, Widened derangements and generalized Laguerre polynomials, Ramanujan J. to appear.

Topic 3: a. Capparelli, Del Fra, Chebyshev coordinates and Salem numbers, submitted

b. Capparelli, Del Fra, Sciò, On the span of polynomials with integer coefficients, Math. Comp. 79 (2010), no. 270, 967-981.

Some papers related to the Capparelli identities:

1. Capparelli, On some representations of twisted affine Lie algebras and combinatorial identities, J. Algebra 154 (1993), no. 2, 335-355

2. Andrews, Schur’s theorem, Capparelli’s conjecture and q-trinomial coefficients, Contemp. Math., 166, Amer. Math. Soc. 1994.

3. Capparelli, A combinatorial proof of a partition identity related to the level 3 representations of a twisted affine Lie algebra, Comm. Algebra 23 (1995) no. 8, 2959-2969

4. Alladi, Andrews, Gordon, Refinements and generalizations of Capparelli’s conjecture on partitions, J. Algebra 174 (1995), no. 2, 636-658

5. Capparelli, A construction of the level 3 modules for the affine Lie algebra A₂⁽²⁾ and a new combinatorial identity of the Rogers-Ramanujan type. Trans. Amer. Math. Soc. 348 (1996), no. 2, 481-501

6. Sills, On series exapansions of Capparelli’s infinite product. Adv. in Appl. Math. 33 (2004), no. 2, 397-408

7. Alladi, Remembering Basil Gordon, Notices AMS, August 2013.

8. Bringmann, Mahlburg False theta functions and companions to Capparelli’s identities. Adv. Math. 278 (2015) 121-136

9. Berkovich, Uncu A new companion to Capparelli’s identities, Adv. in Appl. Math. 71 (2015), 125-137

10. Bringmann, Mahlburg Three-colored partitions and dilated companions of Capparelli’s identities, Analytic number theory, modula forms and q-hypergeometric series, 139-156, Springer Proc. Math. Stat., 221, 2017

11. Fu, Shishuo, Zeng Jiang A unifying combinatorial approach to refined little Göllnitz and Capparelli’s companion identities. Adv. in Appl. Math. 98 (2018), 127-154

12. Dousse, Lovejoy, Generalizations of Capparelli’s identity, arXiv preprint, arxiv:1702.07249, 2017

13. Kursungoz, Andrews-Gordon type series for Capparelli’s and Göllnitz-Gordon identities, arXiv:1807.11189, 2018

14. Alladi, My association and collaboration with George Andrews,

15. Alladi, Andrews, Ono, McIntosh, On the work of Basil Gordon, Journal of Combinatorial Theory, Series A, 113 (2006) 21-38

16. Kanade, Some results on the representation theory of vertex operator algebras and integer partition theory, 2015

17. Kanade, Russell, Staircases to analytic sum-side for many new integer partition identities of Rogers-Ramanujan type, arXiv:1803.032515, 2018

18. Nandi, Partition identities arising from the standard A₂⁽²⁾-modules of level 4,

19. Berkovich, Uncu, Polynomial identities implying Capparelli’s partition theorems, arXiv 1807.10974

20. Dousse, The method of weighted words revisited, arXiv:1702.07177

21. Primc,Sikic: Leading terms of relations for standard modules of affine Lie algebras C_n⁽¹⁾ arXiv:1506.05026

22. Bringmann et al. Proofs and reductions of Kanade Russell’s partition identities, arXiv:1809.06089

23. Sills, A classical q-hypergeometric approach to the A₂⁽²⁾ standard modules, AlladiFest, Florida 2016

24. Berkovich, Uncu, Elementary polynomial identities involving q-trinomial coefficients, arXiv 1810.06497v1, 2018

25. Berkovich, Uncu, Refined q-trinomial coefficients and two infinite hierarchies of q-series identities, arXiv: 1810.12048

26. Dousse, On partition identities of Capparelli and Primc, arXiv:1811.02251

27. Kanade, Searching for modular companions, arXiv:1902.05199

Photos/Memories

Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: quarto

Taken by A. Sills during a visit to the Rutgers Math Department, New Jersey, USA, on 9 April 2004. From left to right: Freeman Dyson (cf. also here), S. C., Jim Lepowsky, George Andrews (cf. also here)