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Rogers Mod 14 Identities

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The Rogers mod 14 identities are a set of three Rogers-Ramanujan-like identities for a complex number q with |q|<1, given by

A(q)=sum_(n=0)^(infty)(q^(n^2))/((q;q)_n(q;q^2)_n)
(1)
=((q^6,q^8,q^(14);q^(14))_infty)/((q;q)_infty)
(2)
=1+q+2q^2+3q^3+5q^4+7q^5+10q^6+14q^7+19q^8+26q^9+...
(3)
B(q)=sum_(n=0)^(infty)(q^(n^2+n))/((q;q)_n(q;q^2)_(n+1))
(4)
=((q^4,q^(10),q^(14);q^(14))_infty)/((q;q)_infty)
(5)
=1+q+2q^2+3q^3+4q^4+6q^5+9q^6+12q^7+17q^8+23q^9+...
(6)
C(q)=sum_(n=0)^(infty)(q^(n^2+2n))/((q;q)_n(q;q^2)_(n+1))
(7)
=((q^2,q^(12),q^(14);q^(14))_infty)/((q;q)_infty)
(8)
=1+q+q^2+2q^3+3q^4+4q^5+6q^6+8q^7+11q^8+15q^9+...
(9)

(OEIS A105780, A105781, and A105782), where (a;q)_n is the q-Pochhammer symbol and (a,b,c;q)_infty=(a;q)_infty(b;q)_infty(c;q)_infty.

The A-identity was found by Rogers (1894) and appears as formula 61 in the list of Slater (1952). The B- and C-identities were found by Rogers (1917) and appeared as formulas 60 and 59 respectively in Slater (1952).

See also

Rogers-Ramanujan Identities

This entry contributed by Andrew Sills

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References

Mc Laughlin, J.; Sills, A. V.; and Zimmer, P. "Rogers-Ramanujan-Slater Type Identities." Elec. J. Combin., Dynamic Survey DS15, Dec. 15, 2022. https://doi.org/10.37236/36.Rogers, L. J. "Second Memoir on the Expansion of Certain Infinite Products." Proc. London Math. Soc. 25, 318-343, 1894.Rogers, L. J. "On Two Theorems of Combinatory Analysis and Some Allied Identities." Proc. London Math. Soc. 16, 315-336, 1917.Slater, L. J. "Further Identities of the Rogers-Ramanujan Type." Proc. London Math. Soc. Ser. 2 54, 147-167, 1952.Sloane, N. J. A. Sequences A105780, A105781, and A105782 in "The On-Line Encyclopedia of Integer Sequences."

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Rogers Mod 14 Identities

Cite this as:

Weisstein, Eric W., with contributions by Andrew Sills. "Rogers Mod 14 Identities." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RogersMod14Identities.html

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