Balanced derivatives, identities, and bounds for trigonometric and Bessel series

Bruce C. Berndt, Martino Fassina, Sun Kim, Alexandru Zaharescu

Research output: Contribution to journalArticlepeer-review


Motivated by two identities published with Ramanujan's lost notebook and connected, respectively, with the Gauss circle problem and the Dirichlet divisor problem, in an earlier paper, three of the present authors derived representations for certain sums of products of trigonometric functions as double series of Bessel functions [8]. These series are generalized in the present paper by introducing the novel notion of balanced derivatives, leading to further theorems. As we will see below, the regions of convergence in the unbalanced case are entirely different than those in the balanced case. From this viewpoint, it is remarkable that Ramanujan had the intuition to formulate entries that are, in our new terminology, “balanced”. If x denotes the number of products of the trigonometric functions appearing in our sums, in addition to proving the identities mentioned above, theorems and conjectures for upper and lower bounds for the sums as x→∞ are established.

Original languageEnglish (US)
Article number108085
JournalAdvances in Mathematics
StatePublished - Feb 24 2022


  • Balanced derivatives
  • Bessel functions
  • Dirichlet divisor problem
  • Ramanujan's lost notebook
  • Trigonometric series

ASJC Scopus subject areas

  • General Mathematics


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