Frobenius’ identity

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11: 24.5 Recurrence Relations

… ►

§24.5(ii) Other Identities

… ►

§24.5(iii) Inversion Formulas

►In each of (24.5.9) and (24.5.10) the first identity implies the second one and vice-versa. …

12: 15.17 Mathematical Applications

… ►

§15.17(iv) Combinatorics

►In combinatorics, hypergeometric identities classify single sums of products of binomial coefficients. …

13: 27.15 Chinese Remainder Theorem

… ►The Chinese remainder theorem states that a system of congruences

x\equiv a_{1}\pmod{m_{1}},\dots,x\equiv a_{k}\pmod{m_{k}}

, always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod

m

), where

m

is the product of the moduli. …

14: 17.17 Physical Applications

… ►In exactly solved models in statistical mechanics (Baxter (1981, 1982)) the methods and identities of §17.12 play a substantial role. …

15: 17.18 Methods of Computation

… ►Lehner (1941) uses Method (2) in connection with the Rogers–Ramanujan identities. …

16: 35.10 Methods of Computation

… ►Koev and Edelman (2006) utilizes combinatorial identities for the zonal polynomials to develop computational algorithms for approximating the series expansion (35.8.1). …

17: 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. …

$x, y$	real variables.
…
$\mathbf{I}$	identity matrix
…

20: 20.11 Generalizations and Analogs

… ►This is the discrete analog of the Poisson identity (§1.8(iv)). … ►In the case

z=0

identities for theta functions become identities in the complex variable

q

, with

\left|q\right|<1

, that involve rational functions, power series, and continued fractions; see Adiga et al. (1985), McKean and Moll (1999, pp. 156–158), and Andrews et al. (1988, §10.7). … ►Similar identities can be constructed for

{{}_{2}F_{1}}\left(\tfrac{1}{3},\tfrac{2}{3};1;k^{2}\right)

{{}_{2}F_{1}}\left(\tfrac{1}{4},\tfrac{3}{4};1;k^{2}\right)

, and

{{}_{2}F_{1}}\left(\tfrac{1}{6},\tfrac{5}{6};1;k^{2}\right)

. …