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Jackson-Slater Identity

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The Jackson-Slater identity is the q-series identity of Rogers-Ramanujan-type given by

sum_(k=0)^(infty)(q^(2k^2))/((q)_(2k))=((q,q^7,q^8;q^8)_infty(q^6,q^(10);q^(16))_infty)/((q)_infty)
(1)
=(f(q^3,q^5))/(f(-q^2))
(2)
=1+q^2+q^3+2q^4+2q^5+3q^6+3q^6+5q^7+...
(3)

(OEIS A069910; Leininger and Milne 1999), where (a^i,b^j,...,a^p;q)_infty is extended q-series notation and f(a,b) is a Ramanujan theta function.

The identity in question was actually first published by Jackson (1928) in slightly disguised form as the fifth equation on page 170 in his paper, though this early appearance of this identity is not well-known. It became widely known as equation 39 (and 83) in the collection of identities due to Slater (1952).

See also

q-Series, Rogers-Ramanujan Identities

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References

Jackson, F. H. "Examples of a Generalization of Euler's Transformation for Power Series." Messenger Math. 57, 169-187, 1928.Leininger, V. E. and Milne, S. C. "Some New Infinite Families of eta-Function Identities." Methods Appl. Anal. 6, 225-248, 1999.Mc Laughlin, J.; Sills, A. V.; and Zimmer, P. "Dynamic Survey DS15: Rogers-Ramanujan-Slater Type Identities." Electronic J. Combinatorics, DS15, 1-59, May 31, 2008. http://www.combinatorics.org/Surveys/ds15.pdf.Slater, L. J. "Further Identities of the Rogers-Ramanujan Type." Proc. London Math. Soc. Ser. 2 54, 147-167, 1952.Sloane, N. J. A. Sequence A069910 in "The On-Line Encyclopedia of Integer Sequences."

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Jackson-Slater Identity

Cite this as:

Weisstein, Eric W. "Jackson-Slater Identity." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Jackson-SlaterIdentity.html

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