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Bailey's Lemma

If, for n>=0,

beta_n=sum_(r=0)^n(alpha_r)/((q;q)_(n-r)(aq;q)_(n+r)),
(1)

then

beta_n^'=sum_(r=0)^n(alpha_r^')/((q;q)_(n-r)(aq;q)_(n+r)),
(2)

where

alpha_r^'=((rho_1;q)_r(rho_2;q)_r(aq/rho_1rho_2)^ralpha_r)/((aq/rho_1;q)_r(aq/rho_2;q)_r)
(3)
beta_n^'=sum_(j>=0)((rho_1;q)_j(rho_2;q)_j(aq/rho1_1rho_2;q)_(n-j)(aq/rho_1rho_2)^jbeta_j)/((q;q)_(n-j)(aq/rho_1;q)_n(aq/rho_2;q)_n).
(4)

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References

Andrews, G. E. "Multiple Series Rogers-Ramanujan Type Identities." Pacific J. Math. 114, 267-283, 1984.Andrews, G. E. "Bailey's Lemma" and "Bailey's Lemma in Computer Algebra." §3.4 and 10.4 in q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. Providence, RI: Amer. Math. Soc., pp. 25-27 and 99-100, 1986.Andrews, G. E. "The Fifth and Seventh Order Mock Theta Functions." Trans. Amer. Soc. 293, 113-134, 1986.Andrews, G. E. "Mock Theta Functions." Proc. Sympos. Pure Math. 49, 283-298, 1989.Andrews, G. E. and Hickerson, D. "Ramanujan's 'Lost' Notebook VII: The Sixth Order Mock Theta Functions." Adv. Math. 89, 60-105, 1991.Bailey, W. N. "Identities of the Rogers-Ramanujan Type." Proc. London Math. Soc. 50, 1-10, 1949.

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Bailey's Lemma

Cite this as:

Weisstein, Eric W. "Bailey's Lemma." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BaileysLemma.html

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