# Rogers–Fine identity

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##### 1: 17.18 Methods of Computation
Lehner (1941) uses Method (2) in connection with the Rogers–Ramanujan identities. …
##### 2: David M. Bressoud
His books are Analytic and Combinatorial Generalizations of the Rogers-Ramanujan Identities, published in Memoirs of the American Mathematical Society 24, No. …
##### 3: 17.12 Bailey Pairs
The Bailey pair that implies the Rogers–Ramanujan identities §17.2(vi) is: …
##### 4: 17.6 ${{}_{2}\phi_{1}}$ Function
###### Fine’s Third Transformation
17.6.11 $\frac{1-z}{1-b}{{}_{2}\phi_{1}}\left({q,aq\atop bq};q,z\right)=\sum_{n=0}^{% \infty}\frac{\left(aq;q\right)_{n}\left(azq/b;q\right)_{2n}b^{n}}{\left(zq,aq/% b;q\right)_{n}}-aq\sum_{n=0}^{\infty}\frac{\left(aq;q\right)_{n}\left(azq/b;q% \right)_{2n+1}(bq)^{n}}{\left(zq;q\right)_{n}\left(aq/b;q\right)_{n+1}},$ $|z|<1,|b|<1$.
##### 6: 26.10 Integer Partitions: Other Restrictions
The set $\{n\geq 1\>|\>n\equiv\pm j\ \pmod{k}\}$ is denoted by $A_{j,k}$. …
###### §26.10(iv) Identities
Equations (26.10.13) and (26.10.14) are the Rogers–Ramanujan identities. …
##### 7: Bibliography R
• J. Riordan (1979) Combinatorial Identities. Robert E. Krieger Publishing Co., Huntington, NY.
• M. D. Rogers (2005) Partial fractions expansions and identities for products of Bessel functions. J. Math. Phys. 46 (4), pp. 043509–1–043509–18.
• ##### 8: 17.16 Mathematical Applications
More recent applications are given in Gasper and Rahman (2004, Chapter 8) and Fine (1988, Chapters 1 and 2).
##### 10: 17.2 Calculus
###### §17.2(vi) Rogers–Ramanujan Identities
These identities are the first in a large collection of similar results. …