Jackson q Bessel function - Alchetron, the free social encyclopedia

In mathematics, a Jackson q-Bessel function (or basic Bessel function) is one of the three q-analogs of the Bessel function introduced by Jackson (1903, 1903b, 1905, 1905b). The third Jackson q-Bessel function is the same as the Hahn–Exton q-Bessel function.

Definition

The three Jackson q-Bessel functions are given in terms of the Pochhammer symbol and the basic hypergeometric function φ by

J ν ( 1 ) ( x ; q ) = ( q ν + 1 ; q ) ∞ ( q ; q ) ∞ ( x / 2 ) ν 2 ϕ 1 ( 0 , 0 ; q ν + 1 ; q , − x 2 / 4 ) J ν ( 2 ) ( x ; q ) = ( q ν + 1 ; q ) ∞ ( q ; q ) ∞ ( x / 2 ) ν 0 ϕ 1 ( ; q ν + 1 ; q , − x 2 q ν + 1 / 4 ) J ν ( 3 ) ( x ; q ) = ( q ν + 1 ; q ) ∞ ( q ; q ) ∞ ( x / 2 ) ν 1 ϕ 1 ( 0 ; q ν + 1 ; q , q x 2 / 4 )

References

Jackson q-Bessel function Wikipedia

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