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On a Generalization of the Pentagonal Number Theorem

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We study a generalization of the classical Pentagonal Number Theorem and its applications. We derive new identities for certain in nite series, recurrence relations and convolution sums for certain restricted partitions and divisor sums. We also derive new identities for Bell polynomials.

Abstract

We study a generalization of the classical Pentagonal Number Theorem and its applications. We derive new identities for certain in nite series, recurrence relations and convolution sums for certain restricted partitions and divisor sums. We also derive new identities for Bell polynomials.

참고문헌 (12)

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  2. J. Bell, "A summary of Euler's work on the pentagonal number theorem", Arch. Hist. Exact Sci, 64(3), pp.301~373 (2010)
  3. S. Bouroubi, "A new identify for complete Bell polynomials based on a formula of Ramanujan", J. Integer Seq, 12, pp.6~ (2009)
  4. T.R. Chandrupatla, "Surprising connections between parti-tions and divisors", College Math. J, 38(4), pp.278~287 (2007)
  5. W.-S. Chaou, "Application of Faadi Bruno's formula in characterization of inverse relations", J. Comput. Appl. Math, 190, pp.151~169 (2006)
  6. P. Cheung, "Quantum calculus, Universitext" (2002), Springer-Verlag
  7. L. Comtet, "Advanced combinatorics: The art of finite and infinite expansions" (1974), D. Reidel Publishing Co
  8. H.-H. Leung, "Another identity for complete Bell polynomials based on Ramanujan's congruences", J. Integer Seq, 21, pp.6~ (2018)
  9. S. Ramanujan, "Proof of certain identities in combinatory analysis", Cambr. Phil. Soc. Proc, 19, pp.211~216 (1919)
  10. J. Riordan, "Combinatorial identities" (1979), Robert E. Krieger Publishing Co
  11. L.J. Rogers, "Third memoir on the expansion of certain infinite products", Proc. London Math. Soc, 26, pp.15~32 (1894)
  12. W. Wang, "General identities on Bell polynomials", Comput. Math. Appl, 58(1), pp.104~118 (2009)

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