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Chu–Vandermonde identity

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1: 17.6 ϕ 1 2 Function
First q -ChuVandermonde Sum
Second q -ChuVandermonde Sum
Rogers–Fine Identity
Nonterminating Form of the q -Vandermonde Sum
2: 15.4 Special Cases
ChuVandermonde Identity
3: Bibliography S
  • L. Shen (1998) On an identity of Ramanujan based on the hypergeometric series F 1 2 ( 1 3 , 2 3 ; 1 2 ; x ) . J. Number Theory 69 (2), pp. 125–134.
  • R. Sitaramachandrarao and B. Davis (1986) Some identities involving the Riemann zeta function. II. Indian J. Pure Appl. Math. 17 (10), pp. 1175–1186.
  • 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.
  • 4: 1.3 Determinants
    If two rows (columns) of a determinant are identical, then the determinant is zero. …
    Vandermonde Determinant or Vandermondian
    5: Preface
    Chu, A. …
    6: 24.10 Arithmetic Properties
    where m n 0 ( mod p - 1 ) . …valid when m n ( mod ( p - 1 ) p ) and n 0 ( mod p - 1 ) , where ( 0 ) is a fixed integer. …
    24.10.8 N 2 n 0 ( mod p ) ,
    valid for fixed integers ( 1 ) , and for all n ( 1 ) such that 2 n 0 ( mod p - 1 ) and p | 2 n .
    24.10.9 E 2 n { 0 ( mod p ) if  p 1 ( mod 4 ) , 2 ( mod p ) if  p 3 ( mod 4 ) ,
    7: Bibliography D
  • J. Dougall (1907) On Vandermonde’s theorem, and some more general expansions. Proc. Edinburgh Math. Soc. 25, pp. 114–132.
  • B. I. Dunlap and B. R. Judd (1975) Novel identities for simple n - j symbols. J. Mathematical Phys. 16, pp. 318–319.
  • 8: 36.9 Integral Identities
    §36.9 Integral Identities
    36.9.9 | Ψ ( E ) ( x , y , z ) | 2 = 8 π 2 3 2 / 3 0 0 2 π ( Ai ( 1 3 1 / 3 ( x + i y + 2 z u exp ( i θ ) + 3 u 2 exp ( - 2 i θ ) ) ) Bi ( 1 3 1 / 3 ( x - i y + 2 z u exp ( - i θ ) + 3 u 2 exp ( 2 i θ ) ) ) ) u d u d θ .
    9: 26.21 Tables
    Andrews (1976) contains tables of the number of unrestricted partitions, partitions into odd parts, partitions into parts ± 2 ( mod 5 ) , partitions into parts ± 1 ( mod 5 ) , and unrestricted plane partitions up to 100. …
    10: 27.16 Cryptography
    Thus, y x r ( mod n ) and 1 y < n . … By the Euler–Fermat theorem (27.2.8), x ϕ ( n ) 1 ( mod n ) ; hence x t ϕ ( n ) 1 ( mod n ) . But y s x r s x 1 + t ϕ ( n ) x ( mod n ) , so y s is the same as x modulo n . …