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    Rogers–Ramanujan continued fraction

    Continued fraction closely related to the Rogers–Ramanujan identities

    The Rogers–Ramanujan continued fraction is a continued fraction discovered by Rogers (1894) and independently by Srinivasa Ramanujan, and closely related to the Rogers–Ramanujan identities.

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    • It can be evaluated explicitly for a broad class of values of its argument.

    Definition

    Given the functions and appearing in the Rogers–Ramanujan identities, and assume ,

    and,

    with the coefficients of the q-expansion being OEIS: A003114 and OEIS: A003106, respectively, where denotes the infinite q-Pochhammer symbol, j is the j-function, and 2F1 is the hypergeometric function.

    The Rogers–Ramanujan continued fraction is then

    is the Jacobi symbol.

    One should be careful with notation since the formulas employing the j-function will be consistent with the other formulas only if (the square of the nome) is used throughout this section since