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On generalized Bessel-Maitland function
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DOI：10.1186/s13662-021-03577-5.png)
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Abstract
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An approach to the generalized Bessel-Maitland function is proposed in the present paper. It is denoted by J(nu,lambda)(mu), where mu > 0 and lambda,nu is an element of C get increasing interest from both theoretical mathematicians and applied scientists. The main objective is to establish the integral representation of J(lambda mu)(nu), by applying Gauss's multiplication theorem and the representation for the beta function as well as Mellin-Barnes representation using the residue theorem. Moreover, the mth derivative of J (lambda)(mu,nu) is considered, and it turns out that it is expressed as the Fox-Wright function. In addition, the recurrence formulae and other identities involving the derivatives are derived. Finally, the monotonicity of the ratio between two modified Bessel-Maitland functions I-nu,lambda(mu) defined by I-nu,lambda(mu)(z) = i(-2 lambda-nu)J(nu,lambda) (mu) (iz) of a different order, the ratio between modified Bessel-Maitland and hyperbolic functions, and some monotonicity results for I-nu,lambda(mu)(z) are obtained where the main idea of the proofs comes from the monotonicity of the quotient of two Maclaurin series. As an application, some inequalities (like Turan-type inequalities and their reverse) are proved. Further investigations on this function are underway and will be reported in a forthcoming paper.
Keywords:
Generalized Bessel-Maitland and Struve functions
Integral representation
Mellin-Barnes integral representation
Differential properties
Monotonicity
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7.4K
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