MONOTONICITY PROPERTIES OF THE GENERALIZED STRUVE FUNCTIONS
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This paper introduces and studies a generalization of the classical Struve function of order $p$ given by \[{}_a\mathtt{S}_{p, c}(x):= \sum_{k=0}^\infty \frac{(-c)^k}{\mathrm{\Gamma}{\left( a k +p+\frac{3}{2}\right)} \mathrm{\Gamma}{\left( k+\frac{3}{2}\right)}} \left(\frac{x}{2}\right)^{2k+p+1}.\] Representation formulae are derived for ${}_a\mathtt{S}_{p, c}.$ Further the function ${}_a\mathtt{S}_{p, c}$ is shown to be a solution of an $(a+1)$-order differential equation. Monotonicity and log-convexity properties for the generalized Struve function ${}_a\mathtt{S}_{p, c}$ are investigated, particulary for the case $c=-1$. As a consequence, Tur\'an-type inequalities are established. For $a=2$ and $c=-1,$ dominant and subordinant functions are obtained for the Struve function ${}_2\mathtt{S}_{p, -1}.$
Abstract
This paper introduces and studies a generalization of the classical Struve function of order $p$ given by \[{}_a\mathtt{S}_{p, c}(x):= \sum_{k=0}^\infty \frac{(-c)^k}{\mathrm{\Gamma}{\left( a k +p+\frac{3}{2}\right)} \mathrm{\Gamma}{\left( k+\frac{3}{2}\right)}} \left(\frac{x}{2}\right)^{2k+p+1}.\] Representation formulae are derived for ${}_a\mathtt{S}_{p, c}.$ Further the function ${}_a\mathtt{S}_{p, c}$ is shown to be a solution of an $(a+1)$-order differential equation. Monotonicity and log-convexity properties for the generalized Struve function ${}_a\mathtt{S}_{p, c}$ are investigated, particulary for the case $c=-1$. As a consequence, Tur\'an-type inequalities are established. For $a=2$ and $c=-1,$ dominant and subordinant functions are obtained for the Struve function ${}_2\mathtt{S}_{p, -1}.$
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