Differential equation and inequalities of the generalized k-Bessel functions

In this paper, we introduce and study a generalization of the k-Bessel function of order ν given by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathtt{W}^{\mathtt{k}}_{\nu , c}(x):= \sum_{r=0}^{\infty } \frac{(-c)^{r}}{\Gamma_{\mathtt{k}} ( r \mathtt{k} +\nu +\mathtt{k} ) r!} \biggl( \frac{x}{2} \biggr) ^{2r+\frac{\nu }{\mathtt{k}}}. $$\end{document}Wν,ck(x):=∑r=0∞(−c)rΓk(rk+ν+k)r!(x2)2r+νk. We also indicate some representation formulae for the function introduced. Further, we show that the function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathtt{W}^{ \mathtt{k}}_{\nu , c}$\end{document}Wν,ck is a solution of a second-order differential equation. We investigate monotonicity and log-convexity properties of the generalized k-Bessel function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathtt{W}^{\mathtt{k}} _{\nu , c}$\end{document}Wν,ck, particularly, in the case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$c=-1$\end{document}c=−1. We establish several inequalities, including a Turán-type inequality. We propose an open problem regarding the pattern of the zeroes of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathtt{W}^{ \mathtt{k}}_{\nu , c}$\end{document}Wν,ck.

for Re(x) > 0. Several properties of the 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 k := k / k are studied in [28]. These functions have the series representation , (1.2) where γ 1 is the Euler-Mascheroni constant. A calculation yields 2 , k > 0 and t > 0. (1.3) Clearly, k is increasing on (0, ∞). The Bessel function of order p given by is a particular solution of the Bessel differential equation (1.7) 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 , p, b, c ∈ C. (1.8) In this paper, we consider the function defined by the series where k > 0, ν > -1, and c ∈ R. As k → 1, the k-Bessel function W 1 ν,1 is reduced to the classical Bessel function J ν , whereas W 1 ν,-1 coincides with the modified Bessel function I ν . Thus, we call the function W k ν,c the generalized k-Bessel function. Basic properties of the k-Bessel and related functions can be found in recent works [8,[19][20][21].
Turán [30] proved that the Legendre polynomials P n (x) satisfy the determinantal inequality where n = 0, 1, 2, . . . , and the equality occurs only for x = ±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]. The aim of this paper is to investigate the influence of the 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 k instead of 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 W k ν,c . More importantly, the function W k ν,c 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 k = 1 and for particular values of c. At the end of Section 2, we give two types of integral representations for W k ν,c . Section 3 is devoted to the investigation of monotonicity and log-convexity properties of the functions W k ν,c 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 W k ν,c , leading to an open problem for future studies.

Representations for the k-Bessel function 2.1 The k-Bessel differential equation
In this section, we find differential equations corresponding to the functions W k ν,c .

Proposition 2.1
Let k > 0 and ν > -k. Then the function W k ν,c is a solution of the homogeneous differential equation Proof Differentiating both sides of (1.9) with respect to x, it follows that This implies (2.2) Now differentiating (2.2) with respect to x and then using the property k (z + k) = z k (z) of the k-gamma function yield A further simplification leads to the differential equation (2.1).

Recurrence relations
Thus we have the difference equation This gives us the second difference equation Thus (2.3) and (2.4) lead to the following recurrence relations.
From definition (1.9) it is clear that The derivative of (2.13) with respect to x is Similarly, Identity (2.9) can be proved by using mathematical induction on m. Recall that 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. Assume that identity (2.9) also holds for any m = r ≥ 2, that is, This implies, for m = r + 1, Hence, identity (2.9) is concluded by the mathematical induction on m.

Integral representations of k-Bessel functions
Now we will derive two integral representations of the functions W k ν,c . For this purpose, we need to recall the k-Beta functions from [15]. The k version of the beta functions is defined by According to [15], we have the identity k (kx) = k x-1 (x). This gives Now (1.9) and (2.18) together yield the first integral representation For the second integral representation, substitute x = r + k/2 and y = ν + k/2 into (2.16). Then (2.17) can be rewritten as Again, the identity k (kx) = k x-1 (x) yields k r + 1 2 k = k r- Finally, for c = ±α 2 , α ∈ R, representation (2.23) respectively leads to Example 2.1 If ν = k/2, then from (2.24) computations give the relation between sine and generalized k-Bessel functions by Similarly, the relation can be derived from (2.25).

Monotonicity and log-convexity properties
This section is devoted to discuss the monotonicity and log-convexity properties of the modified k-Bessel function W k ν,-1 = I k ν . As consequences of those results, we derive several functional inequalities for I k ν . The following result of Biernacki and Krzyż [7] will be required. 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 Then we have the following properties. (a) If ν ≥ μ > -k, then the function increasing on (-k, ∞), that is, for ν ≥ μ > -k, for any fixed x > 0 and k > 0. (c) The function ν → I k ν (x) is decreasing and log-convex on (-k, ∞) for each fixed x > 0.
Proof (a) From (3.1) it follows that Denote w r := f r (ν)/f r (μ). Then . Now, using the property k (y + k) = y k (y), we can show that for all ν ≥ μ > -k. Hence, conclusion (a) follows from the Lemma 3.1.
It now follows from (2.8) that whence I k ν+k /I k ν is increasing for ν > -k and for some fixed x > 0, which concludes (b). (c) It is clear that, for all ν > -k, A logarithmic differentiation of f r (ν) with respect to ν yields since k are increasing functions on (-k, ∞). This implies that f r (ν) is decreasing. Thus, for μ ≥ ν > -k, it follows that which is equivalent to say that the function ν → I k ν is decreasing on (-k, ∞) for some fixed x > 0.
The twice logarithmic differentiation of f r (ν) yields for all k > 0 and ν > -k. Since, a sum of log-convex functions is log-convex, it follows that ν → I k ν is log-convex on (-k, ∞) for each fixed x > 0.
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) → R be a positive integrable Inequality (3.6) is reversed if f and g are monotonic in the opposite sense.

Conclusion
It is shown that the generalized k-Bessel functions W k ν,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 I k ν is decreasing and log-convex on (-k, ∞) for each fixed x > 0. Several other inequalities, especially the Turán-type inequality and reverse Turán-type inequality for I k ν are established.
Furthermore, we investigate the pattern for zeroes of W k,1 ν 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 W k ν,1 are increasing in in both cases. However, we have no any analytical proof for this monotonicity of the zeroes of W k ν,1 . As there are several works on the zeroes of the classical Bessel functions,