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Chu–Vandermonde sums (first and second)

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1: 17.6 ϕ 1 2 Function
First q -ChuVandermonde Sum
Second q -ChuVandermonde Sum
Heine’s First Transformation
Fine’s First Transformation
Nonterminating Form of the q -Vandermonde Sum
2: 15.4 Special Cases
ChuVandermonde Identity
Dougall’s Bilateral Sum
15.4.25 n = - Γ ( a + n ) Γ ( b + n ) Γ ( c + n ) Γ ( d + n ) = π 2 sin ( π a ) sin ( π b ) Γ ( c + d - a - b - 1 ) Γ ( c - a ) Γ ( d - a ) Γ ( c - b ) Γ ( d - b ) .
3: Bibliography S
  • J. A. Stratton, P. M. Morse, L. J. Chu, and R. A. Hutner (1941) Elliptic Cylinder and Spheroidal Wave Functions, Including Tables of Separation Constants and Coefficients. John Wiley and Sons, Inc., New York.
  • J. A. Stratton, P. M. Morse, L. J. Chu, J. D. C. Little, and F. J. Corbató (1956) Spheroidal Wave Functions: Including Tables of Separation Constants and Coefficients. Technology Press of M. I. T. and John Wiley & Sons, Inc., New York.
  • R. Szmytkowski (2009) On the derivative of the associated Legendre function of the first kind of integer degree with respect to its order (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J. Math. Chem. 46 (1), pp. 231–260.
  • R. Szmytkowski (2011) On the derivative of the associated Legendre function of the first kind of integer order with respect to its degree (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J. Math. Chem. 49 (7), pp. 1436–1477.
  • R. Szmytkowski (2012) On parameter derivatives of the associated Legendre function of the first kind (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J. Math. Anal. Appl. 386 (1), pp. 332–342.
  • 4: 1.3 Determinants
    1.3.4 det [ a j k ] = = 1 n a j A j .
    1.3.9 det [ a j k ] 2 ( k = 1 n a 1 k 2 ) ( k = 1 n a 2 k 2 ) ( k = 1 n a n k 2 ) .
    for every distinct pair of j , k , or when one of the factors k = 1 n a j k 2 vanishes. …
    Vandermonde Determinant or Vandermondian
    1.3.19 j , k = - | a j , k - δ j , k |
    5: Bibliography T
  • L. N. Trefethen and D. Bau (1997) Numerical Linear Algebra. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
  • 6: 26.8 Set Partitions: Stirling Numbers
    §26.8(i) Definitions
    §26.8(ii) Generating Functions
    §26.8(iv) Recurrence Relations
    §26.8(v) Identities
    §26.8(vii) Asymptotic Approximations
    7: Preface
    Chu, A. …
    8: Errata
  • Chapters 1 Algebraic and Analytic Methods, 10 Bessel Functions, 14 Legendre and Related Functions, 18 Orthogonal Polynomials, 29 Lamé Functions

    Over the preceding two months, the subscript parameters of the Ferrers and Legendre functions, P n , Q n , P n , Q n , Q n and the Laguerre polynomial, L n , were incorrectly displayed as superscripts.

    Reported by Roy Hughes on 2022-05-23

  • Equation (10.19.11)
    10.19.11 Q 3 ( a ) = 549 28000 a 8 - 1 10767 6 93000 a 5 + 79 12375 a 2

    Originally the first term on the right-hand side of this equation was written incorrectly as - 549 28000 a 8 .

    Reported 2015-03-16 by Svante Janson.

  • 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.

  • Paragraph Case III: V ( x ) = - 1 2 x 2 + 1 4 β x 4 (in §22.19(ii))

    Two corrections have been made in this paragraph. First, the correct range of the initial displacement a is 1 / β | a | < 2 / β . Previously it was 1 / β | a | 2 / β . Second, the correct period of the oscillations is 2 K ( k ) / η . Previously it was given incorrectly as 4 K ( k ) / η .

    Reported 2014-05-02 by Svante Janson.

  • Subsection 14.2(ii)

    Originally it was stated, incorrectly, that Q ν μ ( x ) is real when ν , μ and x ( 1 , ) . This statement is true only for P ν μ ( x ) and Q ν μ ( x ) .

    Reported 2012-07-18 by Hans Volkmer and Howard Cohl.

  • 9: 14.18 Sums
    §14.18 Sums
    §14.18(iii) Other Sums
    14.18.7 ( x - y ) k = 0 n ( 2 k + 1 ) P k ( x ) Q k ( y ) = ( n + 1 ) ( P n + 1 ( x ) Q n ( y ) - P n ( x ) Q n + 1 ( y ) ) - 1 .
    For collections of sums involving associated Legendre functions, see Hansen (1975, pp. 367–377, 457–460, and 475), Erdélyi et al. (1953a, §3.10), Gradshteyn and Ryzhik (2000, §8.92), Magnus et al. (1966, pp. 178–184), and Prudnikov et al. (1990, §§5.2, 6.5). …
    10: 10.31 Power Series
    10.31.1 K n ( z ) = 1 2 ( 1 2 z ) - n k = 0 n - 1 ( n - k - 1 ) ! k ! ( - 1 4 z 2 ) k + ( - 1 ) n + 1 ln ( 1 2 z ) I n ( z ) + ( - 1 ) n 1 2 ( 1 2 z ) n k = 0 ( ψ ( k + 1 ) + ψ ( n + k + 1 ) ) ( 1 4 z 2 ) k k ! ( n + k ) ! ,
    10.31.2 K 0 ( z ) = - ( ln ( 1 2 z ) + γ ) I 0 ( z ) + 1 4 z 2 ( 1 ! ) 2 + ( 1 + 1 2 ) ( 1 4 z 2 ) 2 ( 2 ! ) 2 + ( 1 + 1 2 + 1 3 ) ( 1 4 z 2 ) 3 ( 3 ! ) 2 + .
    10.31.3 I ν ( z ) I μ ( z ) = ( 1 2 z ) ν + μ k = 0 ( ν + μ + k + 1 ) k ( 1 4 z 2 ) k k ! Γ ( ν + k + 1 ) Γ ( μ + k + 1 ) .