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Bessel functions describe the radial part of vibrations of a circular membrane.. Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions y(x) of Bessel's differential equation + + = for an arbitrary complex number, which represents the order of the Bessel function.
Many families of special functions satisfy a recurrence relation that relates the values of the functions of different orders with common argument . The modified Bessel functions of the first kind () satisfy the recurrence relation
By the recurrence relation of Jackson q-Bessel functions and the definition of modified q-Bessel functions, the following recurrence relation can be obtained (() (;) also satisfies the same relation) (Ismail (1981)):
the Bessel-Clifford function evaluated at n=3 divided by 22 as C(3 divided 22,z) from -2-2i to 2+2i. In mathematical analysis, the Bessel–Clifford function, named after Friedrich Bessel and William Kingdon Clifford, is an entire function of two complex variables that can be used to provide an alternative development of the theory of Bessel functions.
One of the advantage of defining this type incomplete-version of Bessel function (,) is that even for example the associated Anger–Weber function defined in Digital Library of Mathematical Functions [6] can related:
In mathematics, the Bessel polynomials are an orthogonal sequence of polynomials. There are a number of different but closely related definitions. There are a number of different but closely related definitions.
In mathematics, the Hahn–Exton q-Bessel function or the third Jackson q-Bessel function is a q-analog of the Bessel function, and satisfies the Hahn-Exton q-difference equation (Swarttouw ). This function was introduced by Hahn ( 1953 ) in a special case and by Exton ( 1983 ) in general.
He developed a new continued fraction technique for calculating the ratios of spherical Bessel functions of consecutive order. This method was an improvement compared to other methods because it started from the beginning of the continued fraction rather than the tail, had a built-in check for convergence, and was numerically stable.