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1: 17.17 Physical Applications
See Kassel (1995). …
2: 17.3 q -Elementary and q -Special Functions
q -Exponential Functions
17.3.1 e q ( x ) = n = 0 ( 1 - q ) n x n ( q ; q ) n = 1 ( ( 1 - q ) x ; q ) ,
17.3.2 E q ( x ) = n = 0 ( 1 - q ) n q ( n 2 ) x n ( q ; q ) n = ( - ( 1 - q ) x ; q ) .
17.3.3 sin q ( x ) = 1 2 i ( e q ( i x ) - e q ( - i x ) ) = n = 0 ( 1 - q ) 2 n + 1 ( - 1 ) n x 2 n + 1 ( q ; q ) 2 n + 1 ,
17.3.4 Sin q ( x ) = 1 2 i ( E q ( i x ) - E q ( - i x ) ) = n = 0 ( 1 - q ) 2 n + 1 q n ( 2 n + 1 ) ( - 1 ) n x 2 n + 1 ( q ; q ) 2 n + 1 .
3: 17.18 Methods of Computation
For computation of the q -exponential function see Gabutti and Allasia (2008). …
4: 20.11 Generalizations and Analogs
With the substitutions a = q e 2 i z , b = q e - 2 i z , with q = e i π τ , we have …
5: 28.14 Fourier Series
28.14.1 me ν ( z , q ) = m = - c 2 m ν ( q ) e i ( ν + 2 m ) z ,
6: 22.11 Fourier and Hyperbolic Series
If q exp ( 2 | ζ | ) < 1 , then … Next, if q exp ( | ζ | ) < 1 , then … Next, with E = E ( k ) denoting the complete elliptic integral of the second kind (§19.2(ii)) and q exp ( 2 | ζ | ) < 1 , …
7: 18.28 Askey–Wilson Class
18.28.3 2 π sin θ w ( cos θ ) = | ( e 2 i θ ; q ) ( a e i θ , b e i θ , c e i θ , d e i θ ; q ) | 2 ,
18.28.8 1 2 π 0 π Q n ( cos θ ; a , b | q ) Q m ( cos θ ; a , b | q ) | ( e 2 i θ ; q ) ( a e i θ , b e i θ ; q ) | 2 d θ = δ n , m ( q n + 1 , a b q n ; q ) , a , b or a = b ¯ ; | a b | < 1 ; | a | , | b | 1 .
18.28.13 C n ( cos θ ; β | q ) = = 0 n ( β ; q ) ( β ; q ) n - ( q ; q ) ( q ; q ) n - e i ( n - 2 ) θ = ( β ; q ) n ( q ; q ) n e i n θ ϕ 1 2 ( q - n , β β - 1 q 1 - n ; q , β - 1 q e - 2 i θ ) .
18.28.15 1 2 π 0 π C n ( cos θ ; β | q ) C m ( cos θ ; β | q ) | ( e 2 i θ ; q ) ( β e 2 i θ ; q ) | 2 d θ = ( β , β q ; q ) ( β 2 , q ; q ) ( 1 - β ) ( β 2 ; q ) n ( 1 - β q n ) ( q ; q ) n δ n , m , - 1 < β < 1 .
8: Bibliography O
  • A. B. Olde Daalhuis (1994) Asymptotic expansions for q -gamma, q -exponential, and q -Bessel functions. J. Math. Anal. Appl. 186 (3), pp. 896–913.
  • 9: 18.27 q -Hahn Class