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Chu–Vandermonde identity

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11: 22.9 Cyclic Identities
§22.9 Cyclic Identities
§22.9(ii) Typical Identities of Rank 2
§22.9(iii) Typical Identities of Rank 3
12: Bibliography T
  • L. N. Trefethen and D. Bau (1997) Numerical Linear Algebra. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
  • 13: 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. …
    14: 15.17 Mathematical Applications
    §15.17(iv) Combinatorics
    In combinatorics, hypergeometric identities classify single sums of products of binomial coefficients. …
    15: 27.15 Chinese Remainder Theorem
    The Chinese remainder theorem states that a system of congruences x a 1 ( mod m 1 ) , , x a k ( mod 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. …
    16: 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. …
    17: 17.18 Methods of Computation
    Lehner (1941) uses Method (2) in connection with the Rogers–Ramanujan identities. …
    18: 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). …
    19: 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. …
    20: Errata
  • Equation (25.15.6)
    25.15.6 G ( χ ) r = 1 k 1 χ ( r ) e 2 π i r / k .

    The upper-index of the finite sum which originally was k , was replaced with k 1 since χ ( k ) = 0 .

    Reported by Gergő Nemes on 2021-08-23

  • Section 1.13

    In Equation (1.13.4), the determinant form of the two-argument Wronskian

    1.13.4 𝒲 { w 1 ( z ) , w 2 ( z ) } = det [ w 1 ( z ) w 2 ( z ) w 1 ( z ) w 2 ( z ) ] = w 1 ( z ) w 2 ( z ) w 2 ( z ) w 1 ( z )

    was added as an equality. In ¶Wronskian (in §1.13(i)), immediately below Equation (1.13.4), a sentence was added indicating that in general the n -argument Wronskian is given by 𝒲 { w 1 ( z ) , , w n ( z ) } = det [ w k ( j 1 ) ( z ) ] , where 1 j , k n . Immediately below Equation (1.13.4), a sentence was added giving the definition of the n -argument Wronskian. It is explained just above (1.13.5) that this equation is often referred to as Abel’s identity. Immediately below Equation (1.13.5), a sentence was added explaining how it generalizes for n th-order differential equations. A reference to Ince (1926, §5.2) was added.

  • Notation

    The overloaded operator is now more clearly separated (and linked) to two distinct cases: equivalence by definition (in §§1.4(ii), 1.4(v), 2.7(i), 2.10(iv), 3.1(i), 3.1(iv), 4.18, 9.18(ii), 9.18(vi), 9.18(vi), 18.2(iv), 20.2(iii), 20.7(vi), 23.20(ii), 25.10(i), 26.15, 31.17(i)); and modular equivalence (in §§24.10(i), 24.10(ii), 24.10(iii), 24.10(iv), 24.15(iii), 24.19(ii), 26.14(i), 26.21, 27.2(i), 27.8, 27.9, 27.11, 27.12, 27.14(v), 27.14(vi), 27.15, 27.16, 27.19).

  • Subsection 5.2(iii)

    Three new identities for Pochhammer’s symbol (5.2.6)–(5.2.8) have been added at the end of this subsection.

    Suggested by Tom Koornwinder.

  • Equation (9.6.26)
    9.6.26 Bi ( z ) = 3 1 / 6 Γ ( 1 3 ) e ζ F 1 1 ( 1 6 ; 1 3 ; 2 ζ ) + 3 7 / 6 2 7 / 3 Γ ( 2 3 ) ζ 4 / 3 e ζ F 1 1 ( 7 6 ; 7 3 ; 2 ζ )

    Originally the second occurrence of the function F 1 1 was given incorrectly as F 1 1 ( 7 6 ; 7 3 ; ζ ) .

    Reported 2014-05-21 by Hanyou Chu.