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Rogers-Ramanujan Functions, Modular Functions, and Computer Algebra

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Advances in Computer Algebra (WWCA 2016)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 226))

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Abstract

Many generating functions for partitions of numbers are strongly related to modular functions. This article introduces such connections using the Rogers-Ramanujan functions as key players. After exemplifying basic notions of partition theory and modular functions in tutorial manner, relations of modular functions to q-holonomic functions and sequences are discussed. Special emphasis is put on supplementing the ideas presented with concrete computer algebra. Despite intended as a tutorial, owing to the algorithmic focus the presentation might contain aspects of interest also to the expert. One major application concerns an algorithmic derivation of Felix Klein’s classical icosahedral equation.

Dedicated to the symbolic summation pioneer Sergei Abramov who concretely passed milestone 70

Both authors were supported by grant SFB F50-06 of the Austrian Science Fund (FWF).

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Notes

  1. 1.

    i.e., it does not satisfy a linear recurrence with polynomial coefficients.

  2. 2.

    In our context it is convenient to normalize as in (9) instead of using the version \(\varDelta (\tau ):= (2 \pi )^{12} \eta (\tau )^{24}\).

  3. 3.

    Despite the cosets being assumed to be pairwise different, it may well be that \([\gamma _i \infty ]_\varGamma =[\gamma _j \infty ]_\varGamma \) for \(i\ne j\).

  4. 4.

    Recall, \(T=\left( {0 \atop 1}{ -1 \atop 0}\right) \), \(S=\left( {1 \atop 0}{ 1 \atop 1}\right) \).

  5. 5.

    \(\left( {0 \atop -1}{ 1 \atop 0}\right) \left( {a \atop 2 c'}{ b \atop d}\right) \left( {0 \atop 1}{ -1 \atop 0}\right) =\left( {d \atop -b}{ -2 c' \atop a}\right) \) implies \(\left( {1 \atop 0}{ 2 \atop 1}\right) \in T^{-1} \varGamma _0(2) T\).

  6. 6.

    The \(B_n\) are the Bernoulli numbers; as for \(\varDelta \), also for the Eisenstein series we prefer the normalized versions.

  7. 7.

    By Lemma 4.

  8. 8.

    By Lemma 4.

  9. 9.

    i.e., a commutative ring with 1 which is also a vector space over \(\mathbb {C}\).

  10. 10.

    In fact one can use the observation (75) from Sect. 13.

  11. 11.

    \(\pi ^{-1}(V_M)\) contains \(\frac{a}{c}\) (\(=\gamma \infty \)), and \(\pi ^{-1}(V_M)\setminus \{\frac{a}{c}\}\) is an open disc in \(\mathbb {H}\) tangent to the real line at \(\frac{a}{c}\).

  12. 12.

    Apart from the requirement to be a homeomorphism, a second property one needs to verify is that such \(\varphi \) also satisfy the Riemann surface compatibility conditions; see, e.g., [11, 25].

  13. 13.

    Cf. Example 6.

  14. 14.

    It is important to note that the orbit sets of modular transformations are discrete; i.e., they do not have a limit point.

  15. 15.

    Often one restricts to consider such functions only on a complete set of orbit representatives; for example, in the case of \(\varGamma =\mathrm {SL}_2(\mathbb {Z})\) to \(\{\tau \in \mathbb {H}: -1/2\leqslant \mathrm {Re}(\tau )\leqslant 0 \, \text{ and }\, \mathrm {Im}(\tau ) \geqslant \mathrm {Im}(e^{i \tau })\}\cup \{0< \mathrm {Re}(\tau )<1/2 \, \text{ and }\, \mathrm {Im}(\tau ) > \mathrm {Im}(e^{i \tau }) \}\) .

  16. 16.

    Equivalently, \(\varPsi _2\) has single poles at all the elements of the orbit \([\infty ]_{\varGamma _0(2)}\).

  17. 17.

    i.e., q is transcendental over \(\mathbb {F}\).

  18. 18.

    See, for instance, [19].

  19. 19.

    That \(((q,q)_n a_n)_{n\geqslant 0}\) is q-holonomic is immediate by q-holonomic closure properties.

  20. 20.

    See, for instance, [19].

  21. 21.

    See [27] for more information about such finite versions of the Rogers-Ramanujan identities.

  22. 22.

    F(z) is also a q-hypergeometric series; its summand sequence \(\left( f_k(z)\right) _{k\geqslant 0}\) is q-hypergeometric over \(\mathbb {K}\) with \(\mathbb {K}=\mathbb {Q}(z)(q)\).

  23. 23.

    In case no such order 2 equation exists, one proceeds with incrementing the order by one.

  24. 24.

    An excellent account on convergence questions related to the Rogers-Ramanujan continued fraction is [10].

  25. 25.

    By stereographic projection the rotations of the sphere turn into Möbius transformations \(z\mapsto \frac{a z + b}{c z +d}\) of the complex plane.

  26. 26.

    This is also immediate from the q-expansion (37) of \(R(\tau )\) at \(\infty \).

  27. 27.

    And also taking into account the fact that j has zeros of multiplicity 3 at each element of the orbit \([\omega ]_{\mathrm {SL}_2(\mathbb {Z})}\), and no zero elsewhere; see Example 38.

  28. 28.

    To obtain an explicit form of this expression set, for instance, \(b_{g,h}(N)=0\) on the right side of [8, (24)].

  29. 29.

    Warning: in many texts on Jacobi theta functions \(q=e^{\pi i \tau }\), in contrast to \(q=e^{2\pi i \tau }\) as throughout this article.

  30. 30.

    The first such “folklore theorem” we considered was Theorem 33.

  31. 31.

    In addition, \(\varphi \) is supposed to be compatible with the other charts; see e.g. [25].

  32. 32.

    From modular forms point of view, (77) deals with the case of forms of weight zero only.

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Acknowledgements

A preliminary version of this article has been presented at FELIM 2017 in Limoges, France. Peter Paule thanks Moulay Barkatou and his colleagues for organizing such a wonderful conference. When preparing the final version of this article, Peter Paule was enjoying outstanding hospitality at the Tian** Center for Applied Mathematics (TCAM), China. Sincerest thanks to Bill Chen and his group for all the kindness! We also want to thank the anonymous referee for carefully reading the paper and giving us several suggestions on how to improve our paper.

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  1. Research Institute for Symbolic Computation (RISC), Johannes Kepler University, Altenbergerstraße 69, 4040, Linz, Austria

    Peter Paule & Silviu Radu

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  1. Research Institute for Symbolic Computation, Johannes Kepler University, Linz, Austria

    Carsten Schneider

  2. Department of Physics and Computer Science, Wilfrid Laurier University, Waterloo, Ontario, Canada

    Eugene Zima

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Paule, P., Radu, S. (2018). Rogers-Ramanujan Functions, Modular Functions, and Computer Algebra. In: Schneider, C., Zima, E. (eds) Advances in Computer Algebra. WWCA 2016. Springer Proceedings in Mathematics & Statistics, vol 226. Springer, Cham. https://doi.org/10.1007/978-3-319-73232-9_10

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