1 Introductions
Motivated with the repeated appearance of the expression
in the combinatorics of creation and annihilation operators [13, 14] and the perturbative computation of Feynman integrals (see [12]), a generalization of the well-known Pochhammer symbols is given in [15] as
for all \(\mathtt{k}>0\), calling it the Pochhammer k-symbol. Closely associated functions that have relation with the Pochhammer symbols are the gamma and beta functions. Hence it is useful to recall some facts about the for \(\operatorname {Re}(x)>0\). Several properties of the
$$ x(x + \mathtt{k}) (x + 2\mathtt{k})\cdots \bigl(x + (n -1)\mathtt{k} \bigr) $$
$$ (x)_{n,\mathtt{k}}:=x(x + \mathtt{k}) (x + 2\mathtt{k})\cdots \bigl(x + (n -1) \mathtt{k} \bigr), $$
k-gamma and k-beta functions. The k-gamma function, denoted as \(\Gamma_{\mathtt{k}}\), is studied in [15] and defined by
$$\begin{aligned} \Gamma_{\mathtt{k}}(x):= \int_{0}^{\infty }t^{x-1}e^{-\frac{t^{ \mathtt{k}}}{\mathtt{k}}}\,dt \end{aligned}$$
(1.1)
k-gamma functions and applications in generalizing other related functions like k-beta and k-digamma functions can be found in [15, 27, 28] and references therein.The k-digamma functions defined by \(\Psi_{\mathtt{k}}:= \Gamma_{\mathtt{k}}'/\Gamma_{\mathtt{k}} \) are studied in [28]. These functions have the series representation
where \(\gamma_{1}\) is the Euler–Mascheroni constant.
$$\begin{aligned} \Psi_{\mathtt{k}}(t):=\frac{\log (\mathtt{k})-\gamma_{1}}{\mathtt{k}}- \frac{1}{t} +\sum _{n=1}^{\infty }\frac{t}{n\mathtt{k}(n\mathtt{k}+t)}, \end{aligned}$$
(1.2)
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A calculation yields
Clearly, \(\Psi_{\mathtt{k}}\) is increasing on \((0, \infty )\).
$$\begin{aligned} \Psi_{\mathtt{k}}'(t)=\sum _{n=0}^{\infty } \frac{1}{(n\mathtt{k}+t)^{2}}, \quad \mathtt{k}>0 \mbox{ and } t>0. \end{aligned}$$
(1.3)
The Bessel function of order p given by
is a particular solution of the Bessel differential equation
Here Γ denotes the gamma function. A solution of the modified Bessel equation
is the modified Bessel function
The Bessel function has several generalizations (see, e.g., [9, 10]) and is notably investigated in [1, 17]. In [1], a generalized Bessel function is defined in the complex plane, and sufficient conditions for it to be univalent, starlike, close-to-convex, or convex are obtained. This generalization is given by the power series
$$ \mathtt{J}_{p}(x):= \sum_{k=0}^{\infty } \frac{(-1)^{k}}{\Gamma { ( k +p+1 ) } \Gamma { ( k+1 ) }} \biggl( \frac{x}{2} \biggr) ^{2k+p} $$
(1.4)
$$\begin{aligned} x^{2} y''(x)+ x y'(x)+ \bigl(x^{2}-p^{2} \bigr)y(x)= 0. \end{aligned}$$
(1.5)
$$\begin{aligned} x^{2} y''(x)+ x y'(x)- \bigl(x^{2}+{\nu }^{2} \bigr)y(x)= 0, \end{aligned}$$
(1.6)
$$ \mathtt{I}_{\nu }(x):= \sum_{k=0}^{\infty } \frac{1}{\Gamma { ( k +\nu +1 ) } \Gamma { ( k+1 ) }} \biggl( \frac{x}{2} \biggr) ^{2k+\nu }. $$
(1.7)
$$\begin{aligned} \mathcal{W}_{p, b, c}(z)= \sum_{k=0}^{\infty } \frac{(-c)^{k} ( \frac{z}{2} ) ^{2k+p+1}}{\Gamma ( k+1 ) \Gamma ( k+p+ \frac{b+2}{2} ) },\quad p, b, c \in \mathbb{C}. \end{aligned}$$
(1.8)
In this paper, we consider the function defined by the series
where \(\mathtt{k}>0\), \(\nu >-1\), and \(c \in \mathbb{R}\). As \(\mathtt{k}\to 1\), the
$$\begin{aligned} \mathtt{W}_{\nu , c}^{\mathtt{k}} (x) :=\sum _{r=0}^{\infty }\frac{(-c)^{r} }{ \Gamma_{\mathtt{k}}(r \mathtt{k}+ \nu + \mathtt{k}) r!} \biggl( \frac{x}{2} \biggr) ^{2r+\frac{ \nu }{\mathtt{k}}}, \end{aligned}$$
(1.9)
k-Bessel function \(\mathtt{W}_{ \nu , 1}^{1}\) is reduced to the classical Bessel function \(J_{\nu }\), whereas \(\mathtt{W}_{\nu , -1}^{1}\) coincides with the modified Bessel function \(I_{\nu }\). Thus, we call the function \(\mathtt{W}_{\nu , c} ^{\mathtt{k}}\) the generalized k-Bessel function. Basic properties of the k-Bessel and related functions can be found in recent works [8, 19‐21].Turán [30] proved that the Legendre polynomials \(P_{n}(x)\) satisfy the determinantal inequality
where \(n = 0, 1, 2, \ldots \) , and the equality occurs only for \(x = \pm 1\). The inequalities similar to (1.10) can be found in the literature [2, 3, 5, 11, 16, 25] for several other functions, for example, ultraspherical polynomials, Laguerre and Hermite polynomials, Bessel functions of the first kind, modified Bessel functions, and the polygamma function. Karlin and Szegö [24] named determinants in (1.10) as Turánians. More details about Turánians can be found in [5, 11, 18, 22, 23, 29].
$$\begin{aligned} \biggl\vert \textstyle\begin{array}{@{}c@{\quad }c@{}} P_{n}(x) & P_{n+1}(x) \\ P_{n+1}(x) & P_{n+2}(x) \end{array}\displaystyle \biggr\vert \leq 0, \quad -1 \le x \le 1, \end{aligned}$$
(1.10)
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The aim of this paper is to investigate the influence of the \(\Gamma_{\mathtt{k}}\) functions on the properties of the
k-Bessel function defined in (1.9). It is shown that the properties of the classical Bessel functions can be extended to the k-Bessel functions. Moreover, we investigate the effects of \(\Gamma_{\mathtt{k}}\) instead of Γ on the monotonicity and log-convexity properties and related inequalities of the k-Bessel functions. The outcomes of our investigation are presented as follows.In Section 2, we derive representation formulae and some recurrence relations for \(\mathtt{W}_{\nu , c}^{\mathtt{k}}\). More importantly, the function \(\mathtt{W}_{\nu , c}^{\mathtt{k}}\) is shown to be a solution of a certain differential equation of second order, which contains (1.5) and (1.6) for the particular case \(\mathtt{k}=1\) and for particular values of c. At the end of Section 2, we give two types of integral representations for \(\mathtt{W}_{\nu , c}^{\mathtt{k}}\).
Section 3 is devoted to the investigation of monotonicity and log-convexity properties of the functions \(\mathtt{W}_{\nu , c} ^{\mathtt{k}}\) and to relation between two
k-Bessel functions of different order. As a consequence, we deduce Turán-type inequalities.In Section 4, we give concluding remarks and list two tables for the zeroes of \(\mathtt{W}^{\mathtt{k}}_{\nu , c}\), leading to an open problem for future studies.
2 Representations for the k-Bessel function
2.1 The k-Bessel differential equation
In this section, we find differential equations corresponding to the functions \(\mathtt{W}_{\nu , c}^{\mathtt{k}}\).
Proposition 2.1
Let
\(\mathtt{k}>0\)
and
\(\nu >-k\). Then the function
\(\mathtt{W}_{ \nu , c}^{\mathtt{k}}\)
is a solution of the homogeneous differential equation
$$\begin{aligned} x^{2} \frac{d^{2}y}{dx^{2}}+ x \frac{dy}{dx}+ \frac{1}{\mathtt{k}^{2}} \bigl( c x^{2} \mathtt{k}- {\nu^{2}} {} \bigr) y=0. \end{aligned}$$
(2.1)
Proof
Differentiating both sides of (1.9) with respect to x, it follows that
This implies
Now differentiating (2.2) with respect to x and then using the property \(\Gamma_{\mathtt{k}}(z+\mathtt{k})= z \Gamma_{\mathtt{k}}(z)\) of the A further simplification leads to the differential equation (2.1). □
$$\begin{aligned} \frac{d}{dx}\mathtt{W}_{\nu , c}^{\mathtt{k}} (x) =\sum _{r=0}^{\infty }\frac{(-c)^{r} ( 2r+\frac{\nu }{\mathtt{k}} ) }{ \Gamma_{\mathtt{k}}(r \mathtt{k}+ \nu + \mathtt{k}) r!} \biggl( \frac{x ^{2r+\frac{\nu }{\mathtt{k}}-1}}{2^{2r+\frac{\nu }{\mathtt{k}}}} \biggr) . \end{aligned}$$
$$\begin{aligned} x\frac{d}{dx}\mathtt{W}_{\nu , c}^{\mathtt{k}} (x) = \sum_{r=0}^{ \infty }\frac{(-c)^{r} ( 2r+\frac{\nu }{\mathtt{k}} ) }{ \Gamma_{\mathtt{k}}(r \mathtt{k}+ \nu + \mathtt{k}) r!} \biggl( \frac{x}{2} \biggr) ^{2r+\frac{\nu }{\mathtt{k}}}. \end{aligned}$$
(2.2)
k-gamma function yield
$$\begin{aligned} &x^{2} \frac{d^{2}}{dx^{2}}\mathtt{W}_{\nu , c}^{\mathtt{k}} (x)+ x \frac{d}{dx}\mathtt{W}_{\nu , c}^{\mathtt{k}} (x) \\ &\quad =\sum_{r=0}^{\infty }\frac{(-c)^{r} ( 2r+\frac{\nu }{\mathtt{k}} ) ^{2} }{ \Gamma_{\mathtt{k}}(r \mathtt{k}+ \nu + \mathtt{k}) r!} \biggl( \frac{x}{2} \biggr) ^{2r+\frac{\nu }{\mathtt{k}}} \\ &\quad = \sum_{r=1}^{\infty }\frac{(-c)^{r} 4r ( r+\frac{\nu }{ \mathtt{k}} ) }{ \Gamma_{\mathtt{k}}(r \mathtt{k}+ \nu + \mathtt{k}) r!} \biggl( \frac{x}{2} \biggr) ^{2r+\frac{\nu }{\mathtt{k}}} \\ &\quad \quad {}+ \frac{\nu^{2}}{\mathtt{k}^{2}} \sum _{r=0}^{\infty }\frac{(-c)^{r} }{ \Gamma_{\mathtt{k}}(r \mathtt{k}+ \nu + \mathtt{k}) r!} \biggl( \frac{x}{2} \biggr) ^{2r+\frac{\nu }{\mathtt{k}}} \\ &\quad =\frac{4}{\mathtt{k}} \sum_{r=1}^{\infty } \frac{(-c)^{r} }{ \Gamma_{\mathtt{k}}(r \mathtt{k}+ \nu ) (r-1)!} \biggl( \frac{x}{2} \biggr) ^{2r+\frac{\nu }{\mathtt{k}}} + \frac{\nu^{2}}{\mathtt{k}^{2}} \mathtt{W}_{\nu , c}^{\mathtt{k}} (x) \\ &\quad =- \frac{c x^{2}}{\mathtt{k}}\mathtt{W}_{\nu , c}^{\mathtt{k}} (x) + \frac{\nu^{2}}{\mathtt{k}^{2}}\mathtt{W}_{\nu , c}^{\mathtt{k}} (x). \end{aligned}$$
2.2 Recurrence relations
From (2.2) we have
Thus we have the difference equation
Again, rewrite (2.2) as
This gives us the second difference equation
Thus (2.3) and (2.4) lead to the following recurrence relations.
$$\begin{aligned} x \frac{d}{dx}\mathtt{W}_{\nu , c}^{\mathtt{k}} (x) &= \frac{1}{ \mathtt{k}}\sum_{r=0}^{\infty } \frac{(-c)^{r} ( 2r \mathtt{k} + {\nu } ) }{ \Gamma_{\mathtt{k}}(r \mathtt{k}+ \nu + \mathtt{k}) r!} \biggl( \frac{x}{2} \biggr) ^{2r+\frac{\nu }{\mathtt{k}}} \\ & =\frac{\nu }{\mathtt{k}}\sum_{r=0}^{\infty }\frac{(-c)^{r} }{ \Gamma_{\mathtt{k}}(r \mathtt{k}+ \nu + \mathtt{k}) r!} \biggl( \frac{x}{2} \biggr) ^{2r+\frac{\nu }{\mathtt{k}}} \\ &\quad {}+2 \sum _{r=1}^{\infty }\frac{(-c)^{r}}{ \Gamma_{\mathtt{k}}(r \mathtt{k}+ \nu + \mathtt{k}) (r-1)!} \biggl( \frac{x}{2} \biggr) ^{2r+\frac{\nu }{\mathtt{k}}} \\ & =\frac{\nu }{\mathtt{k}}\mathtt{W}_{\nu , c}^{\mathtt{k}}(x) +2 \sum _{r=0}^{\infty }\frac{(-c)^{r+1}}{ \Gamma_{\mathtt{k}}(r \mathtt{k}+ \nu + 2\mathtt{k}) r!} \biggl( \frac{x}{2} \biggr) ^{2r+2+\frac{ \nu }{\mathtt{k}}} \\ & =\frac{\nu }{\mathtt{k}}\mathtt{W}_{\nu , c}^{\mathtt{k}}(x)- xc \mathtt{W}_{\nu +\mathtt{k}, c}^{\mathtt{k}}(x). \end{aligned}$$
$$\begin{aligned} x \frac{d}{dx}\mathtt{W}_{\nu , c}^{\mathtt{k}} (x)= \frac{\nu }{ \mathtt{k}}\mathtt{W}_{\nu , c}^{\mathtt{k}}(x)- xc \mathtt{W}_{ \nu +\mathtt{k}, c}^{\mathtt{k}}(x). \end{aligned}$$
(2.3)
$$\begin{aligned} x \frac{d}{dx}\mathtt{W}_{\nu , c}^{\mathtt{k}} (x) &= \frac{1}{ \mathtt{k}}\sum_{r=0}^{\infty } \frac{(-c)^{r} ( 2r \mathtt{k} +2 {\nu } ) - \nu }{ \Gamma_{\mathtt{k}}(r \mathtt{k}+ \nu + \mathtt{k}) r!} \biggl( \frac{x}{2} \biggr) ^{2r+\frac{\nu }{\mathtt{k}}} \\ &=- \frac{\nu }{\mathtt{k}}\sum_{r=0}^{\infty }\frac{(-c)^{r}}{\Gamma_{\mathtt{k}}(r \mathtt{k}+ \nu + \mathtt{k}) r!} \biggl( \frac{x}{2} \biggr) ^{2r+\frac{\nu }{\mathtt{k}}} +2\sum _{r=0}^{\infty }\frac{(-c)^{r} ( r \mathtt{k} +{\nu } ) }{ \Gamma_{\mathtt{k}}(r \mathtt{k}+ \nu + \mathtt{k}) r!} \biggl( \frac{x}{2} \biggr) ^{2r+\frac{\nu }{ \mathtt{k}}} \\ &=- \frac{\nu }{\mathtt{k}} \mathtt{W}_{\nu , c}^{\mathtt{k}} (x) + \frac{x}{ \mathtt{k}} \sum_{r=0}^{\infty } \frac{(-c)^{r}}{ \Gamma_{\mathtt{k}}(r \mathtt{k}+ \nu - \mathtt{k} + \mathtt{k}) r!} \biggl( \frac{x}{2} \biggr) ^{2r+\frac{\nu -\mathtt{k}}{\mathtt{k}}} \\ &=- \frac{\nu }{\mathtt{k}} \mathtt{W}_{\nu , c}^{\mathtt{k}} (x) + \frac{x}{ \mathtt{k}} \mathtt{W}_{\nu -\mathtt{k}, c}^{\mathtt{k}} (x). \end{aligned}$$
$$\begin{aligned} x \frac{d}{dx}\mathtt{W}_{\nu , c}^{\mathtt{k}} (x) = \frac{x}{ \mathtt{k}} \mathtt{W}_{\nu -\mathtt{k}, c}^{\mathtt{k}} (x) - \frac{ \nu }{\mathtt{k}} \mathtt{W}_{\nu , c}^{\mathtt{k}} (x). \end{aligned}$$
(2.4)
Proposition 2.2
Let
\(\mathtt{k}>0\)
and
\(\nu > -\mathtt{k}\). Then
$$\begin{aligned}& 2 \nu \mathtt{W}_{\nu , c}^{\mathtt{k}}(x) =x \mathtt{W}_{\nu - \mathtt{k}, c}^{\mathtt{k}} (x) + xc \mathtt{k} \mathtt{W}_{\nu + \mathtt{k}, c}^{\mathtt{k}}(x), \end{aligned}$$
(2.5)
$$\begin{aligned}& \mathtt{W}^{\mathtt{k}}_{\nu -\mathtt{k}, c}(x) =\frac{2}{x} \sum_{r=0} ^{\infty }(-1)^{r} (\nu + 2 r \mathtt{k}) \mathtt{W}^{\mathtt{k}}_{ \nu + 2r \mathtt{k}, c}(x), \end{aligned}$$
(2.6)
$$\begin{aligned}& \frac{d}{dx} \bigl( x^{\frac{\nu }{\mathtt{k}}} \mathtt{W}^{ \mathtt{k}}_{\nu , c}(x) \bigr) =\frac{x^{\frac{\nu }{\mathtt{k}}}}{ \mathtt{k}} \mathtt{W}^{\mathtt{k}}_{\nu -\mathtt{k}, c}(x), \end{aligned}$$
(2.7)
$$\begin{aligned}& \frac{d}{dx} \bigl( x^{-\frac{\nu }{\mathtt{k}}} \mathtt{W}^{ \mathtt{k}}_{\nu , c}(x) \bigr) =-c x^{-\frac{\nu }{\mathtt{k}}} \mathtt{W}^{\mathtt{k}}_{\nu +\mathtt{k}, c}(x), \end{aligned}$$
(2.8)
$$\begin{aligned}& \frac{d^{m}}{dx^{m}} \bigl( \mathtt{W}^{\mathtt{k}}_{\nu , c}(x) \bigr) = \frac{1}{2^{m} \mathtt{k}^{m}} \sum_{n=0}^{m} (-1)^{n} \left ( \textstyle\begin{array}{@{}c@{}} m \\ n \end{array}\displaystyle \right ) c^{n} \mathtt{k}^{n} \mathtt{W}^{\mathtt{k}}_{\nu -m \mathtt{k}+2 n \mathtt{k}, c}(x) \quad \textit{for all } m \in \mathbb{N}. \end{aligned}$$
(2.9)
Proof
Next to establish (2.6), let us rewrite (2.5) as
Now multiply both sides of (2.10) by \(-c \mathtt{k}\) and replace ν by \(\nu +2\mathtt{k}\). Then we have
Similarly, multiplying both sides of (2.10) by \(c^{2} \mathtt{k}^{2}\) and replacing ν by \(\nu +4\mathtt{k}\) give
Continuing and adding them lead to (2.6).
$$\begin{aligned} \mathtt{W}_{\nu -\mathtt{k}, c}^{\mathtt{k}} (x) + c \mathtt{k} \mathtt{W}_{\nu +\mathtt{k}, c}^{\mathtt{k}}(x) = 2 \frac{\nu }{x} \mathtt{W}_{\nu , c}^{\mathtt{k}}(x). \end{aligned}$$
(2.10)
$$\begin{aligned} - c \mathtt{k} \mathtt{W}_{\nu +\mathtt{k}, c}^{\mathtt{k}} (x) -c ^{2} \mathtt{k}^{2} \mathtt{W}_{\nu +3 \mathtt{k}, c}^{\mathtt{k}}(x) = -2 c \mathtt{k} \frac{\nu +2 \mathtt{k}}{x}\mathtt{W}_{\nu +2 \mathtt{k}, c}^{\mathtt{k}}(x). \end{aligned}$$
(2.11)
$$\begin{aligned} c^{2} \mathtt{k}^{2} \mathtt{W}_{\nu +3 \mathtt{k}, c}^{\mathtt{k}} (x) +c^{3} \mathtt{k}^{3} \mathtt{W}_{\nu +5 \mathtt{k}, c}^{\mathtt{k}}(x) = 2 c^{2}\mathtt{k}^{2} \frac{\nu +4 \mathtt{k}}{x} \mathtt{W}_{\nu +4 \mathtt{k}, c}^{\mathtt{k}}(x). \end{aligned}$$
(2.12)
From definition (1.9) it is clear that
The derivative of (2.13) with respect to x is
Similarly,
$$\begin{aligned} x^{\frac{\nu }{\mathtt{k}}}\mathtt{W}_{\nu , c}^{\mathtt{k}} (x) = \sum_{r=0}^{\infty }\frac{(-c)^{r} }{ \Gamma_{\mathtt{k}}(r \mathtt{k}+ \nu + \mathtt{k})2^{2r+\frac{\nu }{\mathtt{k}}} r!} ( x ) ^{2r+\frac{2\nu }{\mathtt{k}}}. \end{aligned}$$
(2.13)
$$\begin{aligned} \frac{d}{dx} \bigl( x^{\frac{\nu }{\mathtt{k}}}\mathtt{W}_{\nu , c}^{ \mathtt{k}} (x) \bigr) &=\sum_{r=0}^{\infty } \frac{(-c)^{r} (2r+\frac{2 \nu }{\mathtt{k}}) }{ \Gamma_{\mathtt{k}}(r \mathtt{k}+ \nu + \mathtt{k})2^{2r+\frac{\nu }{\mathtt{k}}} r!} ( x ) ^{2r+\frac{2 \nu }{\mathtt{k}}-1} \\ &=\frac{x^{\frac{\nu }{\mathtt{k}}}}{\mathtt{k}}\sum_{r=0}^{\infty } \frac{(-c)^{r} }{ \Gamma_{\mathtt{k}}(r \mathtt{k}+ \nu ) r!} \biggl( \frac{x}{2} \biggr) ^{2r+\frac{\nu }{\mathtt{k}}-1} \\ &= \frac{x^{\frac{\nu }{\mathtt{k}}}}{ \mathtt{k}}\mathtt{W}_{\nu -\mathtt{k}, c}^{\mathtt{k}} (x). \end{aligned}$$
$$\begin{aligned} \frac{d}{dx} \bigl( x^{-\frac{\nu }{\mathtt{k}}}\mathtt{W}_{\nu , c} ^{\mathtt{k}} (x) \bigr) &=\sum_{r=1}^{\infty } \frac{(-c)^{r} 2r }{ \Gamma_{\mathtt{k}}(r \mathtt{k}+ \nu + \mathtt{k})2^{2r+\frac{\nu }{ \mathtt{k}}} r!} ( x ) ^{2r-1} \\ &=x^{-\frac{\nu }{\mathtt{k}}}\sum_{r=1}^{\infty } \frac{(-c)^{r} }{ \Gamma_{\mathtt{k}}(r \mathtt{k}+ \nu +\mathtt{k}) (r-1)!} \biggl( \frac{x}{2} \biggr) ^{2r+\frac{\nu }{\mathtt{k}}-1} \\ &=x^{-\frac{\nu }{\mathtt{k}}}\sum_{r=0}^{\infty } \frac{(-c)^{r+1} }{ \Gamma_{\mathtt{k}}(r \mathtt{k}+ \nu +2\mathtt{k}) r!} \biggl( \frac{x}{2} \biggr) ^{2r+\frac{\nu }{\mathtt{k}}+1} \\ & =-c x^{-\frac{\nu }{\mathtt{k}}} \mathtt{W}_{\nu +\mathtt{k}, c}^{\mathtt{k}} (x). \end{aligned}$$
Identity (2.9) can be proved by using mathematical induction on m. Recall that
and
For \(m=1\), the proof of identity (2.9) is equivalent to showing that
This relation can be obtained by simply adding (2.3) and (2.4). Thus, identity (2.9) holds for \(m=1\).
$$ \left ( \textstyle\begin{array}{@{}c@{}} r \\ r \end{array}\displaystyle \right ) =\left ( \textstyle\begin{array}{@{}c@{}} r \\ 0 \end{array}\displaystyle \right ) =1 $$
$$ \left ( \textstyle\begin{array}{@{}c@{}} r \\ n \end{array}\displaystyle \right ) +\left ( \textstyle\begin{array}{@{}c@{}} r \\ n-1 \end{array}\displaystyle \right ) = \left ( \textstyle\begin{array}{@{}c@{}} r+1 \\ n \end{array}\displaystyle \right ) . $$
$$\begin{aligned} 2 \mathtt{k} \frac{d}{dx}\mathtt{W}_{\nu , c}^{\mathtt{k}} (x) &= \mathtt{W}_{\nu -\mathtt{k}, c}^{\mathtt{k}} (x) - c \mathtt{k} \mathtt{W}_{\nu +\mathtt{k}, c}^{\mathtt{k}}(x). \end{aligned}$$
(2.14)
Assume that identity (2.9) also holds for any \(m=r \geq 2\), that is,
This implies, for \(m=r+1\),
Hence, identity (2.9) is concluded by the mathematical induction on m. □
$$\begin{aligned} \frac{d^{r}}{dx^{r}} \bigl( \mathtt{W}^{\mathtt{k}}_{\nu , c}(x) \bigr) &= \frac{1}{2^{m} \mathtt{k}^{r}} \sum_{n=0}^{r} (-1)^{n} \left ( \textstyle\begin{array}{@{}c@{}} r \\ n \end{array}\displaystyle \right ) c^{n} \mathtt{k}^{n} \mathtt{W}^{\mathtt{k}}_{\nu -r \mathtt{k}+2 n \mathtt{k}, c}(x). \end{aligned}$$
$$\begin{aligned} &\frac{d^{r+1}}{dx^{r+1}} \bigl( \mathtt{W}^{\mathtt{k}}_{\nu , c}(x) \bigr) \\ &\quad = \frac{1}{2^{r} \mathtt{k}^{r}} \sum_{n=0}^{r} (-1)^{n} \left ( \textstyle\begin{array}{@{}c@{}} r \\ n \end{array}\displaystyle \right ) c^{n} \mathtt{k}^{n} \frac{d}{dr}\mathtt{W}^{\mathtt{k}}_{ \nu -r \mathtt{k}+2 n \mathtt{k}, c}(x) \\ &\quad = \frac{1}{2^{r+1} \mathtt{k}^{r+1}} \sum_{n=0}^{r} (-1)^{n} \left ( \textstyle\begin{array}{@{}c@{}} r \\ n \end{array}\displaystyle \right ) c^{n} \mathtt{k}^{n} \bigl(\mathtt{W}^{\mathtt{k}}_{\nu -(r+1) \mathtt{k}+2 n \mathtt{k}, c}(x)-c \mathtt{k}\mathtt{W}^{\mathtt{k}} _{\nu -(r-1) \mathtt{k}+2 n \mathtt{k}, c}(x) \bigr) \\ &\quad = \frac{1}{2^{r+1} \mathtt{k}^{r+1}} \sum_{n=0}^{r} (-1)^{n} \left ( \textstyle\begin{array}{@{}c@{}} r \\ n \end{array}\displaystyle \right ) c^{n} \mathtt{k}^{n} \mathtt{W}^{\mathtt{k}}_{\nu -(r+1) \mathtt{k}+2 n \mathtt{k}, c}(x) \\ & \quad\quad {}- \frac{1}{2^{r+1} \mathtt{k}^{r+1}} \sum_{n=0}^{r} (-1)^{n} \left ( \textstyle\begin{array}{@{}c@{}} r \\ n \end{array}\displaystyle \right ) c^{n+1} \mathtt{k}^{n+1}\mathtt{W}^{\mathtt{k}}_{\nu -(r-1) \mathtt{k}+2 n \mathtt{k}, c}(x) \\ &\quad =\frac{1}{2^{r+1} \mathtt{k}^{r+1}} \Biggl[ \mathtt{W}^{\mathtt{k}} _{\nu -(r+1) \mathtt{k}, c}(x) +\sum _{n=1}^{r} (-1)^{r} \left ( \left ( \textstyle\begin{array}{@{}c@{}} r \\ n \end{array}\displaystyle \right ) +\left ( \textstyle\begin{array}{@{}c@{}} r \\ n-1 \end{array}\displaystyle \right ) \right ) \mathtt{W}^{\mathtt{k}}_{\nu -(r+1) \mathtt{k}+2n \mathtt{k}, c}(x) \\ & \quad \quad {} -(-1)^{r} c^{r+1}\mathtt{k}^{r+1} \mathtt{W}^{\mathtt{k}}_{ \nu +(r+1) \mathtt{k}, c}(x) \Biggr] \\ &\quad =\frac{1}{2^{r+1} \mathtt{k}^{r+1}} \Biggl[ \left ( \textstyle\begin{array}{@{}c@{}} r+1 \\ 0 \end{array}\displaystyle \right ) \mathtt{W}^{\mathtt{k}}_{\nu -(r+1) \mathtt{k}, c}(x) \\ &\quad \quad {}+\sum_{n=1}^{r} (-1)^{r} \left (\textstyle\begin{array}{@{}c@{}} r+1 \\ n \end{array}\displaystyle \right ) \mathtt{W}^{\mathtt{k}}_{\nu -(r+1) \mathtt{k}+2n \mathtt{k}, c}(x) \\ & \quad \quad {} +(-1)^{r+1} \left ( \textstyle\begin{array}{@{}c@{}} r+1 \\ r+1 \end{array}\displaystyle \right ) c^{r+1} \mathtt{k}^{r+1}\mathtt{W}^{\mathtt{k}}_{\nu -(r+1) \mathtt{k}+2(r+1) \mathtt{k}, c}(x) \Biggr] \\ &\quad =\frac{1}{2^{r+1} \mathtt{k}^{r+1}} \sum_{n=0}^{r+1} (-1)^{r} \left ( \textstyle\begin{array}{@{}c@{}} r+1 \\ n \end{array}\displaystyle \right ) \mathtt{W}^{\mathtt{k}}_{\nu -(r+1) \mathtt{k}+2n \mathtt{k},c}(x). \end{aligned}$$
2.3 Integral representations of k-Bessel functions
Now we will derive two integral representations of the functions \(\mathtt{W}_{\nu , c}^{\mathtt{k}}\). For this purpose, we need to recall the Substituting t by \(t^{2}\) on the integral in (2.15), it follows that
Let \(x=(r+1) \mathtt{k}\) and \(y=\nu \). Then from (2.15) and (2.16) we have
According to [15], we have the identity \(\Gamma_{\mathtt{k}}( \mathtt{k} x)= \mathtt{k}^{x-1} \Gamma (x)\). This gives
Now (1.9) and (2.18) together yield the first integral representation
where \(\mathcal{W}_{p, b, c}\) is defined in (1.8).
k-Beta functions from [15]. The k version of the beta functions is defined by
$$\begin{aligned} \mathtt{B}_{\mathtt{k}}(x, y)=\frac{ \Gamma_{\mathtt{k}}(x) \Gamma_{\mathtt{k}}(y)}{\Gamma_{\mathtt{k}}(x+y)}= \frac{1}{\mathtt{k}} \int_{0}^{1} t^{\frac{x}{\mathtt{k}}-1}(1-t)^{\frac{y}{ \mathtt{k}}-1}\,dt. \end{aligned}$$
(2.15)
$$\begin{aligned} \mathtt{B}_{\mathtt{k}}(x, y)=\frac{2}{\mathtt{k}} \int_{0}^{1} t^{\frac{2x}{ \mathtt{k}}-1} \bigl(1-t^{2} \bigr)^{\frac{y}{\mathtt{k}}-1}\,dt. \end{aligned}$$
(2.16)
$$\begin{aligned} \frac{ 1}{\Gamma_{\mathtt{k}}(r \mathtt{k}+\nu +\mathtt{k})}=\frac{2}{ \Gamma_{\mathtt{k}}((r+1) \mathtt{k}) \Gamma_{\mathtt{k}}(\nu )} \int _{0}^{1} t^{2r+1} \bigl(1-t^{2} \bigr)^{\frac{\nu }{\mathtt{k}}-1}\,dt. \end{aligned}$$
(2.17)
$$\begin{aligned} \frac{ 1}{\Gamma_{\mathtt{k}}(r \mathtt{k}+\nu +\mathtt{k})}=\frac{2}{ \mathtt{k}^{r}\Gamma (r+1) \Gamma_{\mathtt{k}}(\nu )} \int_{0}^{1} t ^{2r+1} \bigl(1-t^{2} \bigr)^{\frac{\nu }{\mathtt{k}}-1}\,dt. \end{aligned}$$
(2.18)
$$\begin{aligned} \mathtt{W}_{\nu , c}^{\mathtt{k}} (x) & = \frac{2}{\Gamma_{\mathtt{k}}( \nu )} \biggl( \frac{x }{2} \biggr) ^{\frac{\nu }{\mathtt{k}}} \int_{0} ^{1} t \bigl(1-t^{2} \bigr)^{\frac{\nu }{\mathtt{k}}-1} \sum_{r=0}^{\infty } \frac{(-c)^{r} }{ \Gamma (r+1) r!} \biggl( \frac{x t}{2 \sqrt{\mathtt{k}}} \biggr) ^{2r}\,dt \\ &=\frac{2}{\Gamma_{\mathtt{k}}(\nu )} \biggl( \frac{x }{2} \biggr) ^{\frac{ \nu }{\mathtt{k}}} \int_{0}^{1} t \bigl(1-t^{2} \bigr)^{\frac{\nu }{\mathtt{k}}-1} \mathcal{W}_{0, 1, c} \biggl( \frac{x t}{\sqrt{\mathtt{k}}} \biggr)\,dt, \end{aligned}$$
(2.19)
For the second integral representation, substitute \(x= r+\mathtt{k}/2\) and \(y= \nu +\mathtt{k}/2\) into (2.16). Then (2.17) can be rewritten as
Again, the identity \(\Gamma_{\mathtt{k}}(\mathtt{k} x)= \mathtt{k} ^{x-1} \Gamma (x)\) yields
Further, the Legendre duplication formula (see [4, 6])
shows that
This, together with (2.20) and (2.21), reduces the series (1.9) of \(\mathtt{W}^{\mathtt{k}}_{\nu , c}\) to
Finally, for \(c=\pm \alpha^{2}\), \(\alpha \in \mathbb{R}\), representation (2.23) respectively leads to
and
$$\begin{aligned} \frac{ 1}{\Gamma_{\mathtt{k}}(r \mathtt{k}+\nu +\mathtt{k})}=\frac{2}{ \Gamma_{\mathtt{k}} ( ( r+\frac{1}{2} ) \mathtt{k} ) \Gamma_{\mathtt{k}} ( \nu +\frac{\mathtt{k}}{2} ) } \int_{0} ^{1} t^{2r} \bigl(1-t^{2} \bigr)^{\frac{\nu }{\mathtt{k}}-\frac{1}{2}}\,dt. \end{aligned}$$
(2.20)
$$\begin{aligned} \Gamma_{\mathtt{k}} \biggl( \biggl( r+\frac{1}{2} \biggr) \mathtt{k} \biggr) =\mathtt{k}^{r-\frac{1}{2}}\Gamma \biggl( r+\frac{1}{2} \biggr) . \end{aligned}$$
(2.21)
$$ \Gamma {(z)}\Gamma { \biggl( z+ \frac{1}{2} \biggr) }= 2^{1-2z} \sqrt{\pi } \Gamma {(2z)} $$
(2.22)
$$ \Gamma \biggl( r+\frac{1}{2} \biggr) r!= r \Gamma \biggl( r+ \frac{1}{2} \biggr) \Gamma (r)= \frac{\sqrt{\pi } (2r)!}{2^{2r}}. $$
$$\begin{aligned} \mathtt{W}_{\nu , c}^{\mathtt{k}} (x) & = \frac{2\sqrt{\mathtt{k}}}{ \Gamma_{\mathtt{k}} ( \nu +\frac{\mathtt{k}}{2} ) } \biggl( \frac{x }{2} \biggr) ^{\frac{\nu }{\mathtt{k}}} \int_{0}^{1} \bigl(1-t^{2} \bigr)^{\frac{ \nu }{\mathtt{k}}-\frac{1}{2}} \sum_{r=0}^{\infty } \frac{(-c)^{r} }{ \Gamma (r+1) r!} \biggl( \frac{x t}{2 \sqrt{\mathtt{k}}} \biggr) ^{2r}\,dt \\ &=\frac{2\sqrt{\mathtt{k}}}{\sqrt{\pi }\Gamma_{\mathtt{k}} ( \nu +\frac{ \mathtt{k}}{2} ) } \biggl( \frac{x }{2} \biggr) ^{\frac{\nu }{ \mathtt{k}}} \int_{0}^{1} \bigl(1-t^{2} \bigr)^{\frac{\nu }{\mathtt{k}}- \frac{1}{2}} \sum_{r=0}^{\infty } \frac{(-c)^{r} }{ (2r)!} \biggl( \frac{x t}{ \sqrt{\mathtt{k}}} \biggr) ^{2r}\,dt. \end{aligned}$$
(2.23)
$$\begin{aligned} \mathtt{W}_{\nu , \alpha^{2}}^{\mathtt{k}} (x) =\frac{2\sqrt{ \mathtt{k}}}{\sqrt{\pi }\Gamma_{\mathtt{k}} ( \nu +\frac{ \mathtt{k}}{2} ) } \biggl( \frac{x }{2} \biggr) ^{\frac{\nu }{ \mathtt{k}}} \int_{0}^{1} \bigl(1-t^{2} \bigr)^{\frac{\nu }{\mathtt{k}}- \frac{1}{2}} \cos \biggl( \frac{\alpha x t}{ \sqrt{\mathtt{k}}} \biggr)\,dt \end{aligned}$$
(2.24)
$$\begin{aligned} \mathtt{W}_{\nu , -\alpha^{2}}^{\mathtt{k}} (x) =\frac{2\sqrt{ \mathtt{k}}}{\sqrt{\pi }\Gamma_{\mathtt{k}} ( \nu +\frac{ \mathtt{k}}{2} ) } \biggl( \frac{x }{2} \biggr) ^{\frac{\nu }{ \mathtt{k}}} \int_{0}^{1} \bigl(1-t^{2} \bigr)^{\frac{\nu }{\mathtt{k}}- \frac{1}{2}} \cosh \biggl( \frac{\alpha x t}{ \sqrt{\mathtt{k}}} \biggr)\,dt. \end{aligned}$$
(2.25)
Example 2.1
If \(\nu =\mathtt{k}/2\), then from (2.24) computations give the relation between sine and generalized Similarly, the relation
can be derived from (2.25).
k-Bessel functions by
$$ \sin \biggl( \frac{\alpha x}{\sqrt{\mathtt{k}}} \biggr) = \frac{\alpha }{\mathtt{k}}\sqrt{ \frac{\pi x}{2}}\mathtt{W}_{\frac{\nu }{ \mathtt{k}}, \alpha^{2}}^{\mathtt{k}} (x). $$
$$ \sinh \biggl( \frac{\alpha x}{\sqrt{\mathtt{k}}} \biggr) = \frac{\alpha }{\mathtt{k}}\sqrt{ \frac{\pi x}{2}}\mathtt{W}_{\frac{\nu }{ \mathtt{k}}, -\alpha^{2}}^{\mathtt{k}} (x) $$
3 Monotonicity and log-convexity properties
This section is devoted to discuss the monotonicity and log-convexity properties of the modified
k-Bessel function \(\mathtt{W} _{\nu , -1}^{\mathtt{k}}=\mathtt{I}_{\nu }^{\mathtt{k}}\). As consequences of those results, we derive several functional inequalities for \(\mathtt{I}_{\nu }^{\mathtt{k}}\).The following result of Biernacki and Krzyż [7] will be required.
Lemma 3.1
([7])
Consider the power series
\(f(x)=\sum_{k=0} ^{\infty }a_{k} x^{k}\)
and
\(g(x)=\sum_{k=0}^{\infty }b_{k} x^{k}\), where
\(a_{k} \in \mathbb{R}\)
and
\(b_{k} > 0\)
for all k. Further, suppose that both series converge on
\(\vert x \vert < r\). If the sequence
\(\{a_{k}/b_{k} \}_{k\geq 0}\)
is increasing (or decreasing), then the function
\(x \mapsto f(x)/g(x)\)
is also increasing (or decreasing) on
\((0,r)\).
The lemma still holds when both f and g are even or both are odd functions.
We now state and prove our main results in this section. Consider the functions
where
$$\begin{aligned} \mathcal{I}_{\nu }^{\mathtt{k}}(x):= \biggl( \frac{2}{x} \biggr) ^{\frac{ \nu }{\mathtt{k}}}\Gamma_{\mathtt{k}} (\nu +\mathtt{k}) \mathtt{I} _{\nu }^{\mathtt{k}}(x)=\sum_{r=0}^{\infty }f_{r}( \nu ) x^{2r}, \end{aligned}$$
(3.1)
$$\begin{aligned} \begin{aligned} & \mathtt{I}_{\nu }^{\mathtt{k}}(x)= \mathtt{W}_{\nu , -1}^{\mathtt{k}} (x) =\sum_{r=0}^{\infty }\frac{1 }{ \Gamma_{\mathtt{k}}(r \mathtt{k}+ \nu + \mathtt{k}) r!} \biggl( \frac{x}{2} \biggr) ^{2r+\frac{\nu }{ \mathtt{k}}} \quad \mbox{and} \\ &f_{r}(\nu )= \frac{\Gamma_{ \mathtt{k}}{(\nu +\mathtt{k})}}{\Gamma_{\mathtt{k}}{(r{\mathtt{k}}+ \nu +\mathtt{k})}4^{r} r!}. \end{aligned} \end{aligned}$$
(3.2)
Then we have the following properties.
Theorem 3.1
Let
\(\mathtt{k}>0\). The following results are true for the modified
k-Bessel functions: (a)
If
\(\nu \geq \mu >-\mathtt{k}\), then the function
\(x \mapsto {\mathcal{I}_{\mu }^{\mathtt{k}}(x)}/ {\mathcal{I}_{\nu } ^{\mathtt{k}}(x)}\)
is increasing on
\(\mathbb{R}\).
(b)
The function
\(\nu \mapsto \mathcal{I}^{\mathtt{k}}_{\nu + \mathtt{k}}(x) /\mathcal{I}^{\mathtt{k}}_{\nu }(x)\)
is increasing on
\((-\mathtt{k}, \infty )\), that is, for
\(\nu \geq \mu >-\mathtt{k}\),
for any fixed
\(x>0\)
and
\(\mathtt{k}>0\).
$$ \mathcal{I}_{\nu +\mathtt{k}}^{\mathtt{k}}(x)\mathcal{I}_{\mu }^{ \mathtt{k}}(x) \geq \mathcal{I}_{\nu }^{\mathtt{k}}(x)\mathcal{I}_{ \mu +\mathtt{k}}^{\mathtt{k}}(x) $$
(3.3)
(c)
The function
\(\nu \mapsto \mathcal{I}_{\nu }^{\mathtt{k}}(x)\)
is decreasing and log-convex on
\((-\mathtt{k}, \infty )\)
for each fixed
\(x >0\).
Proof
(a) From (3.1) it follows that
Denote \(w_{r}:=f_{r}(\nu )/f_{r}(\mu )\). Then
Now, using the property \(\Gamma_{\mathtt{k}}{(y+\mathtt{k})}=y \Gamma_{\mathtt{k}}{(y)}\), we can show that
for all \(\nu \geq \mu >-\mathtt{k}\). Hence, conclusion (a) follows from the Lemma 3.1.
$$\begin{aligned} \frac{\mathcal{I}^{\mathtt{k}}_{\nu }(x)}{\mathcal{I}^{\mathtt{k}} _{\mu }(x)} =\frac{\sum_{r=0}^{\infty }f_{r}(\nu ) x^{2r}}{\sum_{r=0} ^{\infty }f_{r}(\mu ) x^{2r}}. \end{aligned}$$
$$ w_{r}= \frac{\Gamma_{\mathtt{k}}{(\nu +\mathtt{k})}\Gamma_{\mathtt{k}} {(r{\mathtt{k}}+\mu +\mathtt{k})}}{\Gamma_{\mathtt{k}}{(\mu + \mathtt{k})} \Gamma_{\mathtt{k}}{(r{\mathtt{k}}+\nu +\mathtt{k})}}. $$
$$\begin{aligned} \frac{w_{r+1}}{w_{r}} &= \frac{\Gamma_{\mathtt{k}}{(r{\mathtt{k}}+ \nu +\mathtt{k})}\Gamma_{\mathtt{k}} {(r{\mathtt{k}}+\mu +2\mathtt{k})}}{ \Gamma_{\mathtt{k}}{(r{\mathtt{k}}+\mu +\mathtt{k})} \Gamma_{ \mathtt{k}}{(r{\mathtt{k}}+\nu +2\mathtt{k})}} =\frac{r{\mathtt{k}}+ \mu +\mathtt{k}}{r{\mathtt{k}}+\nu +\mathtt{k}}\leq 1 \end{aligned}$$
(b) Let \(\nu \geq \mu >-\mathtt{k}\). It follows from part (a) that
on \((0,\infty )\). Thus
It now follows from (2.8) that
whence \(\mathcal{I}^{\mathtt{k}}_{\nu +k}/\mathcal{I}^{\mathtt{k}} _{\nu }\) is increasing for \(\nu >-\mathtt{k}\) and for some fixed \(x >0\), which concludes (b).
$$\begin{aligned} \frac{d}{dx} \biggl( \frac{\mathcal{I}^{\mathtt{k}}_{\nu }(x)}{ \mathcal{I}^{\mathtt{k}}_{\mu }(x)} \biggr) \geq 0 \end{aligned}$$
$$\begin{aligned} \bigl( \mathcal{I}^{\mathtt{k}}_{\nu }(x) \bigr) ' \bigl( \mathcal{I} ^{\mathtt{k}}_{\mu }(x) \bigr) - \bigl( \mathcal{I}^{\mathtt{k}}_{ \nu }(x) \bigr) \bigl( \mathcal{I}^{\mathtt{k}}_{\mu }(x) \bigr) ' \geq 0. \end{aligned}$$
(3.4)
$$\begin{aligned} \frac{x}{2} \bigl(\mathcal{I}^{\mathtt{k}}_{\nu +k}(x) \mathcal{I} ^{\mathtt{k}}_{\mu }(x)-\mathcal{I}^{\mathtt{k}}_{\mu +k}(x) \mathcal{I}^{\mathtt{k}}_{\nu }(x) \bigr) \geq 0, \end{aligned}$$
(c) It is clear that, for all \(\nu >-\mathtt{k}\),
A logarithmic differentiation of \(f_{r}(\nu )\) with respect to ν yields
since \(\Psi_{\mathtt{k}}\) are increasing functions on \((-\mathtt{k}, \infty )\). This implies that \(f_{r}(\nu )\) is decreasing.
$$ f_{r}(\nu )= \frac{\Gamma_{\mathtt{k}}{(\nu +\mathtt{k})}}{ \Gamma_{\mathtt{k}}{(r{\mathtt{k}}+\nu +\mathtt{k})}4^{r} r!}>0. $$
$$\begin{aligned} \frac{f_{r}'(\nu )}{f_{r}(\nu )}= \Psi_{\mathtt{k}}(\nu +\mathtt{k})- \Psi_{\mathtt{k}}(r \mathtt{k}+\nu +\mathtt{k})\leq 0 \end{aligned}$$
Thus, for \(\mu \geq \nu >-\mathtt{k}\), it follows that
which is equivalent to say that the function \(\nu \mapsto \mathcal{I} ^{\mathtt{k}}_{\nu }\) is decreasing on \((-\mathtt{k}, \infty )\) for some fixed \(x >0\).
$$\begin{aligned} \sum_{r=0}^{\infty }f_{r}(\nu ) x^{2r} \geq \sum_{r=0}^{\infty }f_{r}( \mu ) x^{2r}, \end{aligned}$$
The twice logarithmic differentiation of \(f_{r}(\nu )\) yields
for all \(\mathtt{k}>0\) and \(\nu > -\mathtt{k}\). Since, a sum of log-convex functions is log-convex, it follows that \(\nu \to \mathcal{I}_{\nu }^{\mathtt{k}}\) is log-convex on \((-\mathtt{k}, \infty )\) for each fixed \(x>0\). □
$$\begin{aligned} \frac{\partial^{2}}{\partial \nu^{2}} (\log \bigl(f_{r}(\nu ) \bigr) &= \Psi_{k}'(\nu +\mathtt{k})-\Psi_{k}'(r \mathtt{k}+\nu +\mathtt{k}) \\ &=\sum_{n=0}^{\infty } \biggl( \frac{1}{(n\mathtt{k}+\nu +\mathtt{k})^{2}} - \frac{1}{(n\mathtt{k}+r\mathtt{k}+\nu +\mathtt{k})^{2}} \biggr) \\ &=\sum_{n=0}^{\infty }\frac{r \mathtt{k}(2n \mathtt{k}+ r \mathtt{k}+2 \nu +2\mathtt{k}) }{(n\mathtt{k}+\nu +\mathtt{k})^{2}(n\mathtt{k}+r \mathtt{k}+\nu +\mathtt{k})^{2}} \geq 0 \end{aligned}$$
Remark 3.1
One of the most significance consequences of the Theorem 3.1 is the Turán-type inequality for the function \(\mathcal{I}_{\nu }^{\mathtt{k}}\). From the definition of log-convexity it follows from Theorem 3.1(c) that
for \(\alpha \in [0,1]\), \(\nu_{1}, \nu_{2} > -\mathtt{k}\), and \(x >0\). For any \(a\in \mathbb{R}\) and \(\nu \geq -k\), by choosing \(\alpha =1/2, \nu_{1}=\nu -a\), and \(\nu_{2}=\nu +a\), this inequality yields the reverse Turán-type inequality
for any \(\nu \geq \vert a \vert -\mathtt{k}\).
$$\begin{aligned} \mathcal{I}_{\alpha \nu_{1}+(1-\alpha )\nu_{2}}^{\mathtt{k}}(x) \leq \bigl( \mathcal{I}_{\nu_{1}}^{\mathtt{k}} \bigr) ^{\alpha }(x) \bigl( \mathcal{I}_{\nu_{2}}^{\mathtt{k}} \bigr) ^{1-\alpha }(x), \end{aligned}$$
$$\begin{aligned} \bigl( \mathcal{I}_{\nu }^{\mathtt{k}}(x) \bigr) ^{2} - \mathcal{I}_{ \nu -\mathtt{a}}^{\mathtt{k}}(x) \mathcal{I}_{\nu +\mathtt{a}}^{ \mathtt{k}}(x) \leq 0 \end{aligned}$$
(3.5)
Our final result is based on the Chebyshev integral inequality [26, p. 40], which states the following: suppose f and g are two integrable functions and monotonic in the same sense (either both decreasing or both increasing). Let \(q: (a, b) \to \mathbb{R}\) be a positive integrable function. Then
Inequality (3.6) is reversed if f and g are monotonic in the opposite sense.
$$\begin{aligned} \biggl( \int_{a}^{b} q(t) f(t)\,dt \biggr) \biggl( \int_{a}^{b} q(t) g(t)\,dt \biggr) \leq \biggl( \int_{a}^{b} q(t)\,dt \biggr) \biggl( \int_{a}^{b} q(t) f(t) g(t)\,dt \biggr) . \end{aligned}$$
(3.6)
The following function is required:
where
$$\begin{aligned} \mathcal{J}_{\nu }^{\mathtt{k}}(x):= \biggl( \frac{2}{x} \biggr) ^{\frac{ \nu }{\mathtt{k}}}\Gamma_{\mathtt{k}} (\nu +\mathtt{k}) \mathtt{J} _{\nu }^{\mathtt{k}}(x)=\sum_{r=0}^{\infty }g_{r}( \nu ) x^{2r}, \end{aligned}$$
(3.7)
$$\begin{aligned} \begin{aligned} &\mathtt{J}_{\nu }^{\mathtt{k}}(x)= \mathtt{W}_{\nu , 1}^{\mathtt{k}} (x) =\sum_{r=0}^{\infty }\frac{(-1)^{r} }{ \Gamma_{\mathtt{k}}(r \mathtt{k}+ \nu + \mathtt{k}) r!} \biggl( \frac{x}{2} \biggr) ^{2r+\frac{ \nu }{\mathtt{k}}} \quad \text{and} \\ &g_{r}(\nu )= \frac{(-1)^{r} \Gamma_{\mathtt{k}}{(\nu +\mathtt{k})}}{\Gamma_{\mathtt{k}}{(r{\mathtt{k}}+ \nu +\mathtt{k})}4^{r} r!}. \end{aligned} \end{aligned}$$
(3.8)
Theorem 3.2
Let
\(\mathtt{k}>0\). Then, for
\(\nu \in (-3\mathtt{k}/4,- \mathtt{k}/2] \cup [\mathtt{k}/2, \infty ) \),
and
Inequalities (3.9) and (3.10) are reversed if
\(\nu \in (-\mathtt{k}/2, \mathtt{k}/2)\).
$$\begin{aligned} \mathcal{I}_{\nu }^{\mathtt{k}}(x)\mathcal{I}_{\nu + \frac{\mathtt{k}}{2}}^{\mathtt{k}}(x) \leq \frac{\sqrt{\mathtt{k}}}{x} \sin \biggl( \frac{x}{\mathtt{k}} \biggr) \mathcal{I}_{2\nu +\frac{\mathtt{k}}{2}}^{\mathtt{k}}(x) \end{aligned}$$
(3.9)
$$\begin{aligned} \mathcal{J}_{\nu }^{\mathtt{k}}(x)\mathcal{J}_{\nu + \frac{\mathtt{k}}{2}}^{\mathtt{k}}(x) \leq \frac{\sqrt{\mathtt{k}}}{x} \sinh \biggl( \frac{x}{\mathtt{k}} \biggr) \mathcal{J}_{2\nu +\frac{\mathtt{k}}{2}}^{\mathtt{k}}(x). \end{aligned}$$
(3.10)
Proof
Define the functions q, f, and g on \([0, 1]\) as
Then, for any \(x \geq 0\),
Since the functions f and g both are decreasing for \(\nu \geq \mathtt{k}/2\) and both are increasing for \(\nu \in (-3\mathtt{k}/4,- \mathtt{k}/2]\), inequality (3.6) yields (3.9). On the other hand, if \(\nu \in (-\mathtt{k}/2, \mathtt{k}/2)\), then the function f is increasing, but g is decreasing, and hence inequality (3.9) is reversed.
$$ q(t)= \cos \biggl( \frac{x t}{\sqrt{\mathtt{k}}} \biggr) , \qquad f(t)= \bigl(1-t ^{2} \bigr)^{\frac{v}{\mathtt{k}}-\frac{1}{2}}, \qquad g(t)= \bigl(1-t^{2} \bigr)^{\frac{v}{ \mathtt{k}}+\frac{1}{2}}. $$
$$\begin{aligned}& \int_{0}^{1} q(t)\,dt = \int_{0}^{1} \cos \biggl( \frac{x t}{\sqrt{ \mathtt{k}}} \biggr)\,dt= \frac{\sqrt{\mathtt{k}}}{x} \sin \biggl( \frac{x }{\sqrt{\mathtt{k}}} \biggr) , \\& \int_{0}^{1} q(t) f(t)\,dt = \int_{0}^{1} \cos \biggl( \frac{x t}{\sqrt{ \mathtt{k}}} \biggr) \bigl(1-t^{2} \bigr)^{\frac{v}{\mathtt{k}}-\frac{1}{2}}\,dt= \mathcal{I}_{\nu }^{\mathtt{k}}(x) \quad \text{if } \nu \geq - \mathtt{k}, \\& \int_{0}^{1} q(t) g(t)\,dt = \int_{0}^{1} \cos \biggl( \frac{x t}{\sqrt{ \mathtt{k}}} \biggr) \bigl(1-t^{2} \bigr)^{\frac{v}{\mathtt{k}}+\frac{1}{2}}\,dt= \mathcal{I}_{\nu +\mathtt{k}}^{\mathtt{k}}(x) \quad \text{if } \nu \geq - 2\mathtt{k}, \\& \int_{0}^{1} q(t) f(t) g(t)\,dt = \int_{0}^{1} \cos \biggl( \frac{x t}{\sqrt{\mathtt{k}}} \biggr) \bigl(1-t^{2} \bigr)^{\frac{2 v}{\mathtt{k}}}\,dt= \mathcal{I}_{2 \nu +\frac{\mathtt{k}}{2}}^{\mathtt{k}}(x) \quad \text{if } \nu \geq - \frac{3\mathtt{k}}{4}. \end{aligned}$$
4 Conclusion
It is shown that the generalized k-Bessel functions \(W^{k}_{\nu ,c}\) are solutions of a second-order differential equation, which for \(k=1\) is reduced to the well-known second-order Bessel differential equation. It is also proved that the generalized modified k-Bessel function \(\mathcal{I}_{\nu }^{\mathtt{k}}\) is decreasing and log-convex on \((-\mathtt{k}, \infty )\) for each fixed \(x >0\). Several other inequalities, especially the Turán-type inequality and reverse Turán-type inequality for \(\mathcal{I}_{\nu }^{\mathtt{k}}\) are established.
Furthermore, we investigate the pattern for zeroes of \(\mathcal{W} _{\nu }^{\mathtt{k}, 1}\) in two ways: (i) with respect to fixed
k and variation of ν and (ii) with respect to fixed ν and variation of k.From the data in Table 1 and Table 2, we can observe that the zeroes of \(\mathtt{W}_{ \nu , 1}^{\mathtt{k}}\) are increasing in in both cases. However, we have no any analytical proof for this monotonicity of the zeroes of \(W^{k}_{ \nu ,1}\). As there are several works on the zeroes of the classical Bessel functions, the zeroes of \(\mathtt{W}_{ \nu , 1}^{\mathtt{k}}\) would be an interesting topic for future investigations. The monotonicity of the zeroes of \(\mathtt{W}_{ \nu , c}^{\mathtt{k}}\) with respect to c and fixed
k, ν will be another open problem for further studies. Table 1
Positive zeroes of \(\mathtt{W}_{ \nu , 1}^{\mathtt{k}}\) for fixed ν and different
kk
| 0.5 | 1 | 1.5 | 2 | 2.5 |
|---|---|---|---|---|---|
ν = −0.4 and c = 1 | |||||
1st zero | 0.662422 | 1.75098 | 2.42334 | 2.95334 | 3.40423 |
2nd zero | 2.96686 | 4.87852 | 6.24148 | 7.3588 | 8.32849 |
3rd zero | 5.2018 | 8.01663 | 10.0812 | 11.7913 | 13.2836 |
ν = 0.5 and c = 1 | |||||
1st zero | 2.70943 | 3.14159 | 3.55493 | 3.93277 | 4.28026 |
2nd zero | 4.96077 | 6.28319 | 7.38858 | 8.35255 | 9.21757 |
3rd zero | 7.19373 | 9.42478 | 11.2315 | 12.7879 | 14.1752 |
Table 2
Positive zeroes of \(\mathtt{W}_{ \nu , 1}^{\mathtt{k}}\) for different ν and
k
ν
| −0.4 | −0.3 | 0 | 0.5 | 1 | 1.5 | 2 | 2.5 |
|---|---|---|---|---|---|---|---|---|
k = 0.5 and c = 1 | ||||||||
1st zero | 0.662422 | 0.97534 | 1.70047 | 2.70943 | 3.63143 | 4.51146 | 5.36577 | 6.20238 |
2nd zero | 2.96686 | 3.21271 | 3.90328 | 4.96077 | 5.95189 | 6.90209 | 7.82393 | 8.72471 |
3rd zero | 5.2018 | 5.43751 | 6.11911 | 7.19373 | 8.21647 | 9.20314 | 10.1629 | 11.1017 |
k = 1 and c = 1 | ||||||||
1st zero | 1.75098 | 1.92285 | 2.40483 | 3.14159 | 3.83171 | 4.49341 | 5.13562 | 5.76346 |
2nd zero | 4.87852 | 5.04213 | 5.52008 | 6.28319 | 7.01559 | 7.72525 | 8.41724 | 9.09501 |
3rd zero | 8.01663 | 8.17785 | 8.65373 | 9.42478 | 10.1735 | 10.9041 | 11.6198 | 12.3229 |
k = 1.5 and c = 1 | ||||||||
1st zero | 2.42334 | 2.55767 | 2.9453 | 3.55493 | 4.13426 | 4.69286 | 5.2362 | 5.76774 |
2nd zero | 6.24148 | 6.37291 | 6.76069 | 7.38858 | 7.9979 | 8.5923 | 9.1744 | 9.74613 |
3rd zero | 10.0812 | 10.2116 | 10.5986 | 11.2315 | 11.8513 | 12.4599 | 13.0587 | 13.6488 |
k = 2 and c = 1 | ||||||||
1st zero | 2.95334 | 3.06754 | 3.40094 | 3.93277 | 4.44288 | 4.93703 | 5.41885 | 5.8908 |
2nd zero | 7.3588 | 7.47176 | 7.80657 | 8.35255 | 8.88577 | 9.40825 | 9.92154 | 10.4269 |
3rd zero | 11.7913 | 11.9037 | 12.2382 | 12.7879 | 13.3286 | 13.8616 | 14.3875 | 14.907 |
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