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q-cosine function

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1: 17.3 q -Elementary and q -Special Functions
q -Cosine Functions
17.3.5 cos q ( x ) = 1 2 ( e q ( i x ) + e q ( - i x ) ) = n = 0 ( 1 - q ) 2 n ( - 1 ) n x 2 n ( q ; q ) 2 n ,
17.3.6 Cos q ( x ) = 1 2 ( E q ( i x ) + E q ( - i x ) ) = n = 0 ( 1 - q ) 2 n q n ( 2 n - 1 ) ( - 1 ) n x 2 n ( q ; q ) 2 n .
2: 28.2 Definitions and Basic Properties
3: 28.20 Definitions and Basic Properties
28.20.1 w ′′ - ( a - 2 q cosh ( 2 z ) ) w = 0 ,
4: 28.14 Fourier Series
28.14.2 ce ν ( z , q ) = m = - c 2 m ν ( q ) cos ( ν + 2 m ) z ,
5: 28.4 Fourier Series
28.4.1 ce 2 n ( z , q ) = m = 0 A 2 m 2 n ( q ) cos 2 m z ,
28.4.2 ce 2 n + 1 ( z , q ) = m = 0 A 2 m + 1 2 n + 1 ( q ) cos ( 2 m + 1 ) z ,
6: 28.6 Expansions for Small q
28.6.21 2 1 / 2 ce 0 ( z , q ) = 1 - 1 2 q cos 2 z + 1 32 q 2 ( cos 4 z - 2 ) - 1 128 q 3 ( 1 9 cos 6 z - 11 cos 2 z ) + ,
28.6.22 ce 1 ( z , q ) = cos z - 1 8 q cos 3 z + 1 128 q 2 ( 2 3 cos 5 z - 2 cos 3 z - cos z ) - 1 1024 q 3 ( 1 9 cos 7 z - 8 9 cos 5 z - 1 3 cos 3 z + 2 cos z ) + ,
7: 28.10 Integral Equations
8: 28.32 Mathematical Applications
28.32.4 2 K z 2 - 2 K ζ 2 = 2 q ( cos ( 2 z ) - cos ( 2 ζ ) ) K .
9: 28.9 Zeros
§28.9 Zeros
For real q each of the functions ce 2 n ( z , q ) , se 2 n + 1 ( z , q ) , ce 2 n + 1 ( z , q ) , and se 2 n + 2 ( z , q ) has exactly n zeros in 0 < z < 1 2 π . They are continuous in q . For q the zeros of ce 2 n ( z , q ) and se 2 n + 1 ( z , q ) approach asymptotically the zeros of He 2 n ( q 1 / 4 ( π - 2 z ) ) , and the zeros of ce 2 n + 1 ( z , q ) and se 2 n + 2 ( z , q ) approach asymptotically the zeros of He 2 n + 1 ( q 1 / 4 ( π - 2 z ) ) . …Furthermore, for q > 0 ce m ( z , q ) and se m ( z , q ) also have purely imaginary zeros that correspond uniquely to the purely imaginary z -zeros of J m ( 2 q cos z ) 10.21(i)), and they are asymptotically equal as q 0 and | z | . …