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Bessel functions describe the radial part of vibrations of a circular membrane.. Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, [1] 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 Fourier–Bessel series of a function f(x) with a domain of [0, b] satisfying f(b) = 0. Bessel function for (i) = and (ii) =.: [,] is the representation of that function as a linear combination of many orthogonal versions of the same Bessel function of the first kind J α, where the argument to each version n is differently scaled, according to [1] [2] ():= (,) where u α,n is a root ...
In mathematics, a recurrence relation is an equation according to which the th term of a sequence of numbers is equal to some combination of the previous terms. Often, only previous terms of the sequence appear in the equation, for a parameter that is independent of ; this number is called the order of the relation.
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.
A Lommel polynomial R m,ν (z) is a polynomial in 1/z giving the recurrence relation + ... where J ν (z) is a Bessel function of the first kind. [1]
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.