---
title: Generalized k-Bessel Function
url: https://www.emergentmind.com/topics/generalized-k-bessel-function
type: topic
---

# Generalized k-Bessel Function

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 $\nu$ and parameter $c \in \mathbb{R}$ is given by the convergent power series
$$
W^{k}_{\nu,c}(x) = \sum_{n=0}^\infty \frac{(-c)^n}{\Gamma_k(n k + \nu + k) n!}\left(\frac{x}{2}\right)^{2n+\nu/k},
$$
which reduces to $J_\nu(x)$ or $I_\nu(x)$ in the limits $k \to 1,\, c= \pm 1$ [1611.07499, 1612.08079].

A further generalization encompasses four parameters $(\gamma, \lambda, b, c)$:
$$
\omega_{k,\nu,b,c}^{\gamma, \lambda}(z) = \sum_{n=0}^\infty 
\frac{(-1)^n c^n\, (\gamma)_{n,k}\, \Gamma_k(\nu + \lambda n + \tfrac{b+1}{2})}
{(n!)^2}
\left(\frac{z}{2}\right)^{\nu + 2n}.
$$
This series recovers the two-parameter $k$-Bessel for suitable choices of parameters, and in the limit $k=1$, $\gamma=1$, $\lambda=1$, $b = c = 1$ reduces to the classical Bessel function $J_\nu(z)$ [1705.04599].

## 2. Analytic Structure and Special Representations

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

- **Hadamard Factorization**: For $k>0$, $c>0$, $v>0$,
  $$
  {}_kW_{v,c}(z) = \frac{(z/2)^{v/k}}{\Gamma_k(v+k)} \prod_{n=1}^\infty \left(1 - \frac{z^2}{\omega_{v,c,n}^2}\right),
  $$
  where $\{\omega_{v,c,n}\}$ are its positive real zeros, all of which are simple and lie strictly between corresponding zeros of its derivative [1902.09979].

- **Wright-type Hypergeometric Representation**:
  $$
  W^{k}_{\nu,c}(x) = \frac{1}{k^{\frac{\nu + k + 1}{k}} \Gamma\left(\frac{\nu + 1}{k}\right)}
  {}_1\Psi_2\left[
    \left(\frac{\nu+1}{k}, 1\right);
    (1, 1), \left(\frac{\nu+1}{k}, \frac{1}{k}\right);
    -\frac{c x^2}{4 k^{1/k}}
  \right]
  $$
  with the generalized Wright function defined by
  $$
  {}_p\Psi_q\left[
      (a_i, \alpha_i)_{1:p}\ ;\ 
      (b_j, \beta_j)_{1:q}\ ;
      z
  \right]
  = \sum_{n=0}^\infty \frac{\prod_{i=1}^p \Gamma(a_i + \alpha_i n)}{\prod_{j=1}^q \Gamma(b_j + \beta_j n)} \frac{z^n}{n!}
  $$
  [1612.08079, 1702.05524].

- **Laplace-type Integrals and Connection to Dunkl Kernels**: For higher-rank root systems (e.g., $B_2$), Laplace-type formulas express the generalized Bessel function as
  $$
  D_k^W(x, y) = \int_{z \in \mathbb{R}^2} e^{\langle x, z \rangle} H_k(y, z) \,dz
  $$
  where $H_k(y, z)$ is supported on the Weyl-orbit hull and given by explicit polynomial-integral formulas in $u, v$ [1611.05696].

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

The function ${}_kW_{v,c}(z)$ belongs to the Laguerre–Pólya class, guaranteeing all zeros are real and simple when $k>0$ and $c>0$. The interlacing of the zeros with those of its derivative underpins the determination of "radii of starlikeness" and "convexity" for normalized analytic transforms of ${}_kW_{v,c}$.

Subclasses of normalized functions,
$$
{}_k f_{v,c}(z), \quad
{}_k g_{v,c}(z), \quad
{}_k h_{v,c}(z)
$$
are constructed so $f(0)=0, f'(0)=1$ in $|z|<1$, facilitating geometric function theory analysis [1902.09979]. For example, the smallest positive $r$ such that
$$
\Re \frac{z f'(z)}{f(z)} > 0
$$
for $|z| < r$ gives the starlikeness radius $r^*(f)$.

Explicit bounds for the radii are achieved using Euler-Rayleigh inequalities applied to the Hadamard product. If $S_m = \sum_n \omega_{v, c, n}^{-2m}$, then
$$
S_1^{-1} < \omega_{v, c, 1}^2 < S_1/S_2
$$
yields sharp two-sided estimates for the first positive zero [1902.09979].

## 4. Integral Equations and Fractional Calculus

Generalized $k$-Bessel functions naturally arise in the context of fractional kinetic equations, such as
$$
N(t) - N_0\, \omega_{k, \mu, b, c}^{\gamma, \lambda}(t) = -d^\nu {}_0 D_t^{-\nu} N(t),
$$
where ${}_0 D_t^{-\nu}$ is the Riemann–Liouville fractional integral. The solution is given in terms of the Mittag–Leffler function,
$$
N(t) = N_0 \sum_{n=0}^\infty 
\frac{(-1)^n c^n (\gamma)_{n, k} \Gamma_k(\mu + \lambda n + \frac{b+1}{2}) \Gamma(\mu + 2n + 1)}
{(n!)^2}
\left(\frac{t}{2}\right)^{\mu + 2n}
E_{\nu,\,\mu + 2n + 1}(-d^\nu t^\nu)
$$
with $E_{\alpha,\beta}(z) = \sum_{r=0}^\infty \frac{z^r}{\Gamma(\alpha r + \beta)}$ [1705.04599].

## 5. Functional Equations, Recurrences, and Inequalities

The generalized $k$-Bessel functions satisfy differential and difference equations generalizing the standard Bessel relations:
- **Differential equation**:
  $$
  y''(x) + \frac{1}{x} y'(x) + [c k - (\nu/k)^2 x^{-2}] y(x) = 0
  $$
  which specializes to the classical Bessel ($k=1, c=1$) and modified Bessel ($k=1, c=-1$) ODEs [1611.07499].
- **Recurrence relations**:
  $$
  2\nu W_\nu = x W_{\nu - k} + c k x W_{\nu + k}
  $$
  with further differentiation formulas expressing $d^m W_\nu/dx^m$ in terms of shifts of $W_{\nu\pm mk + 2 n k}$ [1611.07499].

The function $I^k_\nu(x) = W^k_{\nu, -1}(x)$, the modified $k$-Bessel of the first kind, is subject to refined monotonicity and log-convexity properties. For $\mu > \nu > -k$, $x \mapsto I^k_\mu(x)/I^k_\nu(x)$ is strictly increasing; $\nu \mapsto I^k_\nu(x)$ is log-convex and hence satisfies Turán-type inequalities
$$
[I^k_\nu(x)]^2 \leq I^k_{\nu-a}(x) I^k_{\nu+a}(x),\quad \nu \geq |a| - k
$$
[1611.07499, 1702.05524].

## 6. Integral Formulas and Special Values

Unified integral representations for $k$-Bessel functions are provided in terms of generalized $k$-Wright functions. For instance,
$$
\int_0^1 z^{\lambda - 1} (1-z)^{2\lambda - 1} (1 - z/3)^{2(\lambda+p) - 1} (1 - z/4)^{\lambda - 1} W^k_{v, c} \left( \frac{y}{2} z (1-z/3)^2 \right) dz
$$
has a closed expression involving
$$
{}_1 \Psi_2 \Biggl[ \left( \frac{\lambda}{k} + \frac{v}{k}, 2 \right); \left( \frac{v}{k}+1,1 \right), \left( \frac{\lambda}{k} + p + \frac{v}{k}, 2 \right) ;\ -\frac{c y^2}{4k} \Biggr]
$$
[1612.08079]. By specializations among $(k, \gamma, c)$, this unifies many classical results.

## 7. Generalized Modified Bessel Functions and Further Developments

Two-parameter extensions of the Bessel $K$-function, notably $K_{z, w}(x)$, enrich the analytic theory through series, Mellin–Barnes, and double-integral forms:
$$
K_{z, w}(x) = \frac{1}{2\pi i}
\int_{(\!c)} \Gamma\left(\frac{s-z}{2}\right) \Gamma\left(\frac{s+z}{2}\right) {}_1 F_1 \left(\frac{s-z}{2};\frac{1}{2};-\frac{w^2}{4}\right)
{}_1 F_1 \left(\frac{s+z}{2};\frac{1}{2};-\frac{w^2}{4}\right)
(2x)^{-s} ds
$$
$K_{z, w}(x)$ satisfies $K_{z, w}(x) = K_{-z, w}(x) = K_{z, -w}(x)$ and when $w \to 0$ reduces to the ordinary $K_z$ [1706.05363, 1810.03093]. At $z=1/2$, explicit expansions in Humbert functions $\Phi_3$ are known, leading to applications in modular-type transformation formulae for functions such as the Dedekind $\eta$ [1810.03093].

## References

- [1705.04599] "Certain Fractional Kinetic Equations Involving Generalized k-Bessel Function"
- [1902.09979] "Radii of starlikeness and convexity of generalized $k-$Bessel functions"
- [1612.08079] "Certain unified integration formulas associated with generalized k-Bessel function"
- [1611.05696] "Laplace-type integral representations of the generalized Bessel function and of the Dunkl kernel of type $B_2$"
- [1611.07499] "Representation Formulae and Monotonicity of the Generalized k-Bessel Functions"
- [1702.05524] "Inequalities for the modified k-Bessel function"
- [1601.06487] "Certain unified integral formulas involving the generalized modified k-bessel function of first kind"
- [1706.05363] "A generalized modified Bessel function and a higher level analogue of the theta transformation formula"
- [1810.03093] "The generalized modified Bessel function $K_{z,w}(x)$ at $z=1/2$ and Humbert functions"

Source: https://www.emergentmind.com/topics/generalized-k-bessel-function