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Partition identities arising from theta function identities. (English) Zbl 1154.11036
This paper is devoted to a study of the combinatorics of identities that generally have the form $f(q) + f(-q) = g(q)$ or $f(q) - f(-q) = qh(q),$ where $$f(q)$$ is some infinite product which is essentially a modular function. The motivation is one such identity of H. M. Farkas and I. Kra [Contemp. Math. 251, 197–203 (2000; Zbl 1050.11086)], $(-q;q^2)_{\infty}(-q^7;q^{14})_{\infty} - (q;q^2)_{\infty}(q^7;q^{14})_{\infty} = 2q(-q^2;q^2)_{\infty}(-q^{14};q^{28})_{\infty},$ which, along with its combinatorial interpretation, was popularized and generalized by S. O. Warnaar [J. Comb. Theory, Ser. A 110, No. 1, 43–52 (2005; Zbl 1101.11046)]. Here we have used the usual basic hypergeometric notation.
Of course if $$f(q)$$ is a modular function then both the sum $$f(q) + f(-q)$$ and the difference $$f(q) - f(-q)$$ will both be so as well. The authors identify around 15 cases where this sum or difference is an infinite product and offer combinatorial interpretations in terms of colored partitions. For example, we record their Theorem 3.4:
Let $$A(N)$$ denote the number of partitions of $$2N+1$$ into odd parts that are not multiples of $$3$$, with each having two colors, say orange and blue. Let $$B(N)$$ denote the number of partitions of $$2N$$ into four distinct colors, with two colors, say red and green, appearing at most once and only in multiples of $$2$$, one color, say pink, appearing at most once and only in multiples of $$4$$, and the remaining color, say violet, appearing at most once and only in multiples of $$12$$. Then for $$N \geq 2$$, we have $$A(N) = 2B(N)$$.
This follows from the identity $\begin{split} \frac{1}{(q;q^6)_{\infty}^2(q^5;q^6)_{\infty}^2} - \frac{1}{(-q;q^6)_{\infty}^2(-q^5;q^6)_{\infty}^2} = 4q(-q^2;q^2)_{\infty}^4(-q^4;q^4)_{\infty}(-q^{12};q^{12})_{\infty}.\end{split}$ The authors’ proofs of such identities rely heavily on work of H. Schröter [De aequationibus modularibus, Dissertatio Inauguralis, Albertina Litterarum Universitate, Regiomonti, Königsberg, 1854].

##### MSC:
 11P83 Partitions; congruences and congruential restrictions 05A17 Combinatorial aspects of partitions of integers 33D15 Basic hypergeometric functions in one variable, $${}_r\phi_s$$
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##### References:
  Farkas, H. M., Kra, I.: Partitions and theta constant identities, in The Mathematics of Leon Ehrenpreis, Contemp. Math. No. 251, American Mathematical Society, Providence, RI, 2000, 197–203 · Zbl 1050.11086  Schröter, H.: De aequationibus modularibus, Dissertatio Inauguralis, Albertina Litterarum Universitate, Regiomonti, 1854  Ramanujan, S.: Notebooks (2 volumes), Tata Institute of Fundamental Research, Bombay, 1957 · Zbl 0138.24201  Baruah, N. D., Berndt, B. C.: Partition identities and Ramanujan’s modular equations. J. Combin. Theory Ser. A, 114, 1024–1045 (2007) · Zbl 1206.11132  Berndt, B. C.: Partition-theoretic interpretations of certain modular equations of Schröter, Russell, and Ramanujan. Ann. of Combin., to appear · Zbl 1131.05008  Berndt, B. C.: Ramanujan’s Notebooks, Part III, Springer-Verlag, New York, 1991 · Zbl 0733.11001  Berndt, B. C.: Ramanujan’s Notebooks, Part V, Springer-Verlag, New York, 1998 · Zbl 0886.11001
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