Generalized k-Bessel Function

• Generalized k-Bessel function is defined via a convergent series using k-gamma and k-Pochhammer symbols, unifying standard and modified Bessel functions.
• Its analytic structure features Hadamard factorization and Wright-type hypergeometric representations, leading to sharp estimates for zeros and starlikeness radii.
• Applications in fractional calculus, geometric function theory, and mathematical physics support its use in solving integral equations and kinetic models.

The generalized k-Bessel function is a class of special functions defined through extensions of the classical Bessel function, where the parameter $k>0$ interpolates gamma-type structures and introduces significant additional flexibility. Its various formulations unify and extend standard and modified Bessel functions, admit intricate analytic properties, and play a crucial role in fractional calculus, geometric function theory, and mathematical physics.

1. Basic Definitions and Notation

The foundational ingredients are the $k$-gamma function and its associated $k$-Pochhammer symbol:

• $k$-Gamma function: For $k>0$ and $\gamma \in \mathbb{C}$,

$$\Gamma_k(\gamma) = k^{\frac{\gamma}{k} - 1} \Gamma\left(\frac{\gamma}{k}\right), \quad \text{with}\ \Gamma_k(\gamma + k) = \gamma\, \Gamma_k(\gamma),\ \Gamma_k(k) = 1.$$

• $k$-Pochhammer symbol:

$$(\gamma)_{n,k} = \prod_{j=0}^{n-1} (\gamma + j k) = \frac{\Gamma_k(\gamma + n k)}{\Gamma_k(\gamma)},\quad (\gamma)_{0,k}=1.$$

The “standard” generalized $k$-Bessel function of order $k$0 and parameter $k$1 is given by the convergent power series

$k$2

which reduces to $k$3 or $k$4 in the limits $k$5 (Mondal, 2016, Rahman et al., 2016).

A further generalization encompasses four parameters $k$6:

$k$7

This series recovers the two-parameter $k$8-Bessel for suitable choices of parameters, and in the limit $k$9, $k$0, $k$1, $k$2 reduces to the classical Bessel function $k$3 (Agarwal et al., 2017).

2. Analytic Structure and Special Representations

The generalized $k$4-Bessel function exhibits an entire character, with the following notable representations:

• Hadamard Factorization: For $k$5, $k$6, $k$7,

$k$8

where $k$9 are its positive real zeros, all of which are simple and lie strictly between corresponding zeros of its derivative (Toklu, 2019).

• Wright-type Hypergeometric Representation:

$k$0

with the generalized Wright function defined by

$k$1

(Rahman et al., 2016, Mondal et al., 2017).

• Laplace-type Integrals and Connection to Dunkl Kernels: For higher-rank root systems (e.g., $k$2), Laplace-type formulas express the generalized Bessel function as

$k$3

where $k$4 is supported on the Weyl-orbit hull and given by explicit polynomial-integral formulas in $k$5 (Amri et al., 2016).

3. Geometric Function Theory: Zeros, Starlikeness, and Convexity

The function $k$6 belongs to the Laguerre–Pólya class, guaranteeing all zeros are real and simple when $k$7 and $k$8. The interlacing of the zeros with those of its derivative underpins the determination of "radii of starlikeness" and "convexity" for normalized analytic transforms of $k$9.

Subclasses of normalized functions,

$k>0$0

are constructed so $k>0$1 in $k>0$2, facilitating geometric function theory analysis (Toklu, 2019). For example, the smallest positive $k>0$3 such that

$k>0$4

for $k>0$5 gives the starlikeness radius $k>0$6.

Explicit bounds for the radii are achieved using Euler-Rayleigh inequalities applied to the Hadamard product. If $k>0$7, then

$k>0$8

yields sharp two-sided estimates for the first positive zero (Toklu, 2019).

4. Integral Equations and Fractional Calculus

Generalized $k>0$9-Bessel functions naturally arise in the context of fractional kinetic equations, such as

$\gamma \in \mathbb{C}$0

where $\gamma \in \mathbb{C}$1 is the Riemann–Liouville fractional integral. The solution is given in terms of the Mittag–Leffler function,

$\gamma \in \mathbb{C}$2

with $\gamma \in \mathbb{C}$3 (Agarwal et al., 2017).

5. Functional Equations, Recurrences, and Inequalities

The generalized $\gamma \in \mathbb{C}$4-Bessel functions satisfy differential and difference equations generalizing the standard Bessel relations:

• Differential equation:

$\gamma \in \mathbb{C}$5

which specializes to the classical Bessel ($\gamma \in \mathbb{C}$6) and modified Bessel ($\gamma \in \mathbb{C}$7) ODEs (Mondal, 2016).

• Recurrence relations:

$\gamma \in \mathbb{C}$8

with further differentiation formulas expressing $\gamma \in \mathbb{C}$9 in terms of shifts of $$\Gamma_k(\gamma) = k^{\frac{\gamma}{k} - 1} \Gamma\left(\frac{\gamma}{k}\right), \quad \text{with}\ \Gamma_k(\gamma + k) = \gamma\, \Gamma_k(\gamma),\ \Gamma_k(k) = 1.$$0 (Mondal, 2016).

The function $$\Gamma_k(\gamma) = k^{\frac{\gamma}{k} - 1} \Gamma\left(\frac{\gamma}{k}\right), \quad \text{with}\ \Gamma_k(\gamma + k) = \gamma\, \Gamma_k(\gamma),\ \Gamma_k(k) = 1.$$1, the modified $$\Gamma_k(\gamma) = k^{\frac{\gamma}{k} - 1} \Gamma\left(\frac{\gamma}{k}\right), \quad \text{with}\ \Gamma_k(\gamma + k) = \gamma\, \Gamma_k(\gamma),\ \Gamma_k(k) = 1.$$2-Bessel of the first kind, is subject to refined monotonicity and log-convexity properties. For $$\Gamma_k(\gamma) = k^{\frac{\gamma}{k} - 1} \Gamma\left(\frac{\gamma}{k}\right), \quad \text{with}\ \Gamma_k(\gamma + k) = \gamma\, \Gamma_k(\gamma),\ \Gamma_k(k) = 1.$$3, $$\Gamma_k(\gamma) = k^{\frac{\gamma}{k} - 1} \Gamma\left(\frac{\gamma}{k}\right), \quad \text{with}\ \Gamma_k(\gamma + k) = \gamma\, \Gamma_k(\gamma),\ \Gamma_k(k) = 1.$$4 is strictly increasing; $$\Gamma_k(\gamma) = k^{\frac{\gamma}{k} - 1} \Gamma\left(\frac{\gamma}{k}\right), \quad \text{with}\ \Gamma_k(\gamma + k) = \gamma\, \Gamma_k(\gamma),\ \Gamma_k(k) = 1.$$5 is log-convex and hence satisfies Turán-type inequalities

$$\Gamma_k(\gamma) = k^{\frac{\gamma}{k} - 1} \Gamma\left(\frac{\gamma}{k}\right), \quad \text{with}\ \Gamma_k(\gamma + k) = \gamma\, \Gamma_k(\gamma),\ \Gamma_k(k) = 1.$$6

(Mondal, 2016, Mondal et al., 2017).

6. Integral Formulas and Special Values

Unified integral representations for $$\Gamma_k(\gamma) = k^{\frac{\gamma}{k} - 1} \Gamma\left(\frac{\gamma}{k}\right), \quad \text{with}\ \Gamma_k(\gamma + k) = \gamma\, \Gamma_k(\gamma),\ \Gamma_k(k) = 1.$$7-Bessel functions are provided in terms of generalized $$\Gamma_k(\gamma) = k^{\frac{\gamma}{k} - 1} \Gamma\left(\frac{\gamma}{k}\right), \quad \text{with}\ \Gamma_k(\gamma + k) = \gamma\, \Gamma_k(\gamma),\ \Gamma_k(k) = 1.$$8-Wright functions. For instance,

$$\Gamma_k(\gamma) = k^{\frac{\gamma}{k} - 1} \Gamma\left(\frac{\gamma}{k}\right), \quad \text{with}\ \Gamma_k(\gamma + k) = \gamma\, \Gamma_k(\gamma),\ \Gamma_k(k) = 1.$$9

has a closed expression involving

$k$0

(Rahman et al., 2016). By specializations among $k$1, this unifies many classical results.

7. Generalized Modified Bessel Functions and Further Developments

Two-parameter extensions of the Bessel $k$2-function, notably $k$3, enrich the analytic theory through series, Mellin–Barnes, and double-integral forms:

$k$4

$k$5 satisfies $k$6 and when $k$7 reduces to the ordinary $k$8 (Dixit et al., 2017, Kumar, 2018). At $k$9, explicit expansions in Humbert functions $$(\gamma)_{n,k} = \prod_{j=0}^{n-1} (\gamma + j k) = \frac{\Gamma_k(\gamma + n k)}{\Gamma_k(\gamma)},\quad (\gamma)_{0,k}=1.$$0 are known, leading to applications in modular-type transformation formulae for functions such as the Dedekind $$(\gamma)_{n,k} = \prod_{j=0}^{n-1} (\gamma + j k) = \frac{\Gamma_k(\gamma + n k)}{\Gamma_k(\gamma)},\quad (\gamma)_{0,k}=1.$$1 (Kumar, 2018).

• (Agarwal et al., 2017) "Certain Fractional Kinetic Equations Involving Generalized k-Bessel Function"
• (Toklu, 2019) "Radii of starlikeness and convexity of generalized $$(\gamma)_{n,k} = \prod_{j=0}^{n-1} (\gamma + j k) = \frac{\Gamma_k(\gamma + n k)}{\Gamma_k(\gamma)},\quad (\gamma)_{0,k}=1.$$2Bessel functions"
• (Rahman et al., 2016) "Certain unified integration formulas associated with generalized k-Bessel function"
• (Amri et al., 2016) "Laplace-type integral representations of the generalized Bessel function and of the Dunkl kernel of type $$(\gamma)_{n,k} = \prod_{j=0}^{n-1} (\gamma + j k) = \frac{\Gamma_k(\gamma + n k)}{\Gamma_k(\gamma)},\quad (\gamma)_{0,k}=1.$$3"
• (Mondal, 2016) "Representation Formulae and Monotonicity of the Generalized k-Bessel Functions"
• (Mondal et al., 2017) "Inequalities for the modified k-Bessel function"
• (Nisar et al., 2016) "Certain unified integral formulas involving the generalized modified k-bessel function of first kind"
• (Dixit et al., 2017) "A generalized modified Bessel function and a higher level analogue of the theta transformation formula"
• (Kumar, 2018) "The generalized modified Bessel function $$(\gamma)_{n,k} = \prod_{j=0}^{n-1} (\gamma + j k) = \frac{\Gamma_k(\gamma + n k)}{\Gamma_k(\gamma)},\quad (\gamma)_{0,k}=1.$$4 at $$(\gamma)_{n,k} = \prod_{j=0}^{n-1} (\gamma + j k) = \frac{\Gamma_k(\gamma + n k)}{\Gamma_k(\gamma)},\quad (\gamma)_{0,k}=1.$$5 and Humbert functions"