Jackson q-Bessel function
In mathematics, a Jackson "q"-Bessel function (or basic Bessel function) is one of the three "q"-analogs of the Bessel function introduced by . 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 "q"-Pochhammer symbol and the basic hypergeometric function formula_1 by They can be reduced to the Bessel function by the continuous limit: There is a connection formula between the first and second Jackson "q"-Bessel function (): For integer order, the "q"-Bessel functions satisfy Properties. Negative Integer Order. By using the  relations (): we obtain Zeros. Hahn mentioned that formula_11 has infinitely many real zeros (). Ismail proved that for formula_12 all non-zero roots of formula_11 are real (). Ratio of "q"-Bessel Functions. The function formula_14 is a completely monotonic function (). Recurrence Relations. The first and second Jackson "q"-Bessel function have the following recurrence relations (see and ): Inequalities. When formula_12, the second Jackson "q"-Bessel function satisfies: formula_18 For formula_19, formula_20 Generating Function. The following formulas are the "q"-analog of the generating function for the Bessel function (see ): formula_23 is the "q"-exponential function. Alternative Representations. Integral Representations. The second Jackson "q"-Bessel function has the  following integral representations (see and ): where formula_26is the "q"-Pochhammer symbol. This representation reduces to the integral representation of the Bessel function in the limit formula_27. Hypergeometric Representations. The second Jackson "q"-Bessel function has the following hypergeometric representations (see , ): An asymptotic expansion can be obtained as an immediate consequence of the second formula. For other hypergeometric representations, see . Modified "q"-Bessel Functions. The "q"-analog of the modified Bessel functions are defined with the Jackson "q"-Bessel function ( and ): There is a connection formula between the modified q-Bessel functions: For statistical applications, see . Recurrence Relations. By the recurrence relation of Jackson "q"-Bessel functions  and the definition of modified "q"-Bessel functions, the following recurrence relation can be obtained (formula_35 also satisfies the same relation) (): For other recurrence relations, see . Continued Fraction Representation. The ratio of modified "q"-Bessel functions form a continued fraction (): Alternative Representations. Hypergeometric Representations. The function formula_38 has the following representation (): Integral Representations. The modified "q"-Bessel functions have the following integral representations ():