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Continued fractions for square series generating functions

Abstract

We consider new series expansions for variants of the so-termed ordinary geometric square series generating functions originally defined in the recent article titled “Square Series Generating Function Transformations” (arXiv:1609.02803). Whereas the original square series transformations article adapts known generating function transformations to construct integral representations for these square series functions enumerating the square powers of \(q^{n^2}\) for some fixed non-zero q with \(|q| < 1\), we study the expansions of these special series through power series generated by Jacobi-type continued fractions, or J-fractions. We prove new exact expansions of the hth convergents to these continued fraction series and show that the limiting case of these convergent generating functions exists as \(h \rightarrow \infty \). We also prove new infinite q-series representations of special square series expansions involving square-power terms of the series parameter q, the q-Pochhammer symbol, and double sums over the q-binomial coefficients. Applications of the new results we prove within the article include new q-series representations for the ordinary generating functions of the special sequences, \(r_p(n)\), and \(\sigma _1(n)\), as well as parallels to the examples of the new integral representations for theta functions, series expansions of infinite products and partition function generating functions, and related unilateral special function series cited in the first square series transformations article.

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Fig. 1

Notes

  1. 1.

    Notation: Iverson’s convention compactly specifies boolean-valued conditions and is equivalent to the Kronecker delta function, \(\delta _{i,j}\), as \(\left[ n = k\right] _{\delta } \equiv \delta _{n,k}\). Similarly, \(\left[ \mathtt {cond = True}\right] _{\delta } \equiv \delta _{\mathtt {cond}, \mathtt {True}}\) in the remainder of the article.

  2. 2.

    Additional identities of K. Dilcher from his article titled “Some q-series identities related to divisor functions” provide expansions of the series coefficients of \(\left( q; q\right) _{\infty } \times \sum _n n^{\alpha +1} q^n / \left( q; q\right) _{n}\) when \(\alpha \in \mathbb {Z}^{+}\). We can then generate these series by considering higher-order derivatives of our new J-fraction series for \(1 / \left( z; q\right) _{\infty }\) with respect to z in the known Stirling number transformations proved in [6, §2].

References

  1. 1.

    Flajolet, P.: Combinatorial aspects of continued fractions. Discrete Math. 32, 125–161 (1980)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Flajolet, P.: On congruences and continued fractions for some classical combinatorial quantities. Discrete Math. 41, 145–153 (1982)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers. Oxford University Press, Oxford (2008)

    MATH  Google Scholar 

  4. 4.

    Lando, S.K.: Lectures on Generating Functions. American Mathematical Society, Providence, RI (2002)

    Google Scholar 

  5. 5.

    Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W.: NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge (2010)

    MATH  Google Scholar 

  6. 6.

    Schmidt, M.D.: Square series generating function transformations (2016). (arXiv:1609.02803)

  7. 7.

    Schmidt, M.D.: Jacobi type continued fractions for the ordinary generating functions of generalized factorial functions. J. Integr. Seq. 20 (2017)

  8. 8.

    Wall, H.S.: Analytic Theory of Continued Fractions. Chelsea Publishing Company, Bronx (1948)

    MATH  Google Scholar 

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Correspondence to Maxie D. Schmidt.

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Schmidt, M.D. Continued fractions for square series generating functions. Ramanujan J 46, 795–820 (2018). https://doi.org/10.1007/s11139-017-9971-9

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Keywords

  • Square series
  • q-Series
  • J-fraction
  • Continued fraction
  • Sum of squares functions
  • Sum of divisors function
  • Theta function
  • Ordinary generating function

Mathematics Subject Classification

  • 05A15
  • 11Y65
  • 11B65
  • 40A15