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Rogers-Selberg Identities

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The Rogers-Selberg identities are a set of three analytic q-series identities of Rogers-Ramanujan-type appearing as equations 33, 32, and 31 in Slater (1952). For |q|<1, they are

A(q)=sum_(n=0)^(infty)(q^(2n^2))/((q^2;q^2)_n(-q;q)_(2n))
(1)
=((q^3,q^4,q^7;q^7)_infty)/((q^2;q^2)_infty)
(2)
=1+q^2-q^3+q^4-q^5+2q^6-2q^7+3q^8-...
(3)
B(q)=sum_(n=0)^(infty)(q^(2n^2+2n))/((q^2;q^2)_n(-q;q)_(2n))
(4)
=((q^2,q^5,q^7;q^7)_infty)/((q^2;q^2)_infty)
(5)
=1+q^4-q^5+q^6-q^7+2q^8-2q^9+2q^(10)-...
(6)
C(q)=sum_(n=0)^(infty)(q^(2n^2+2n))/((q^2;q^2)_n(-q;q)_(2n+1))
(7)
=((q,q^6,q^7;q^7)_infty)/((q^2;q^2)_infty)
(8)
=1-q+q^2-q^3+2q^4-2q^5+2q^6-3q^7+...,
(9)

(OEIS A104408, A104409, and A104410), where (a,b,c;q)_infty=(a;q)_infty(b;q)_infty(c;q)_infty uses the q-Pochhammer symbol.

Andrews (1980) gave a technique for combinatorially interpreting the Rogers-Selberg identities.

The identities were discovered by Rogers (1894, 1917) and independently rediscovered by Selberg (1936) and Dyson (1943). They were subsequently generalized by Bailey (1947) before appearing in Slater's list of 130 identities of the Rogers-Ramanujan type (Slater 1952).

See also

Rogers-Ramanujan Identities

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References

Andrews, G. E. "Gap-Frequency Partitions and the Rogers-Selberg Identities." Ars. Combin. 9, 201-210, 1980.Bailey, W. N. "Some Identities in Combinatory Analysis." Proc. London Math. Soc. 49, 421-435, 1947. https://doi.org/10.1112/plms/s2-49.6.421.Dyson, F. J. "Three Identities in Combinatory Analysis." J. London Math. Soc. 18, 35-39, 1943.Gasper, G. and Rahman, M. Basic Hypergeometric Series. Cambridge, England: Cambridge University Press, pp. 36-37, 1990.Hahn, H. "Septic Analogues of the Rogers-Ramanujan Functions." Acta Arith. 110, 381-399, 2003.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.Milne, S. C. "Classical Partition Functions and the U(n+1) Rogers-Selberg Identity." Disc. Math. 99, 199-246, 1992.Rogers, L. J. "Second Memoir on the Expansion of Certain Infinite Products." Proc. London Math. Soc. 25, 318-343, 1894. https://doi.org/10.1112/plms/s1-25.1.318.Rogers, L. J. "On Two Theorems of Combinatory Analysis and Some Allied Identities." Proc. London Math. Soc. 16, 315-336, 1917.Selberg, A. "Über einige arithmetische Identitäten." Avh. Norske Vid.-Akad. Oslo I, No. 8, 1-23, 1936.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 A104408, A104409, and A104410 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Rogers-Selberg Identities

Cite this as:

Weisstein, Eric W. "Rogers-Selberg Identities." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Rogers-SelbergIdentities.html

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