--- title: 'Multi-index Wright Function: Fractional Hyper-Bessel' url: https://www.emergentmind.com/topics/multi-index-wright-function type: topic --- # Multi-index Wright Function: Fractional Hyper-Bessel The **multi-index Wright function** is a multiple-parameter entire function associated with fractional hyper-Bessel operators and Caputo fractional derivatives. In the notation used by Droghei, it is denoted by \(\mathcal W^{(\bar\alpha,\bar\nu)}(z)\), with parameter arrays \(\bar\alpha=(\alpha_1,\dots,\alpha_{n+1})\) and \(\bar\nu=(\nu_1,\dots,\nu_n)\), and it arises from an eigenvalue problem for a multi-order fractional hyper-Bessel operator [2301.04640]. The term is not uniform across the literature: in parallel, the expression “multi-index Wright function” is also used for the generalized Fox–Wright function \({}_p\Psi_q(z)\), whose asymptotic theory was systematized by Wright and Braaksma and revisited numerically by Paris [1708.04824]. ## 1. Defining framework and notation Droghei introduces the notation \[ \bar\alpha=(\alpha_1,\dots,\alpha_{n+1}),\qquad \bar\nu=(\nu_1,\dots,\nu_n), \] with the conventions \(\alpha_0=\nu_0=0\), and defines the auxiliary quantities \[ a_j=1+\sum_{m=1}^j(\nu_{m-1}-\alpha_m),\qquad b_j=1+\sum_{m=1}^j(\nu_{m-1}-\alpha_{m-1}), \] for \(j=1,\dots,n+1\). These satisfy \(a_j=b_j-\alpha_j\). The function \(\mathcal W^{(\bar\alpha,\bar\nu)}(z)\) is given by a power series whose coefficients are products of gamma-ratios built from the shifts \(\alpha_{n+1}i+a_j\), \(\alpha_{n+1}i+b_j\), and the terminal factor involving \(\alpha_{n+1}k+b_{n+1}\) [2301.04640]. A central structural fact is that \(\mathcal W^{(\bar\alpha,\bar\nu)}\) is **entire**. The summary of Droghei’s results states that, since \(\alpha_j>0\), the ratio test yields convergence for all \(z\in\mathbb C\). In the related formulation \(W(a,D;z)\) from the 2021 paper, entireness is again established by the D’Alembert criterion together with Wendel’s asymptotic formula, under \(\Re a_j>0\) and \(\Re \nu_j>0\) [2102.04347]. The three-parameter reduction \[ \mathcal W_{\alpha,\beta,\nu}(x^\beta)=\mathcal W^{((\alpha,\beta),(\nu))}(x^\beta) \] plays a special role throughout the theory. It is the setting in which the closest links to the classical Wright and Mittag-Leffler functions, as well as the parameter-derivative formulas, are written most explicitly [2301.04640]. ## 2. Fractional hyper-Bessel origin The multi-index Wright function was introduced as the natural solution of an eigenvalue problem for a fractional hyper-Bessel type operator involving Caputo derivatives [2102.04347]. In the notation of the 2021 paper, the operator is \[ \mathcal D(a,D)=\Bigl(\prod_{j=1}^n x^{\nu_j}\,{}^C\!D_x^{\alpha_j}\Bigr)\,{}^C\!D_x^{\alpha_{n+1}}, \] where \({}^C\!D_x^\alpha\) denotes the Caputo derivative. Its action on monomials is computed explicitly: \[ \mathcal D(a,D)\bigl[x^{\alpha_{n+1}k}\bigr] = \Bigl(\prod_{s=1}^n(\nu_s-\alpha_s+\alpha_{n+1}k)\Bigr) x^{\sum_{s=1}^n(\nu_s-\alpha_s)+\alpha_{n+1}k}. \] Summing termwise in the defining series yields the operator identity \[ \mathcal D(a,D)\Bigl\{W(a,D;x^{\alpha_{n+1}})\Bigr\} = x^{\sum_{s=1}^n(\nu_s-\alpha_s)}\,W(a,D;x^{\alpha_{n+1}}). \] Hence, whenever \(\sum_{s=1}^n(\nu_s-\alpha_s)=0\), the function becomes an eigenfunction of \(\mathcal D(a,D)\) [2102.04347]. The same paper presents the associated fractional ODE \[ \mathcal D(a,D)\,y(x)=x^{\sum_{s=1}^n(\nu_s-\alpha_s)}\,y(x), \] with initial condition \(y(0)=1\) and all lower Caputo derivatives at \(0\) equal to zero. The power-series solution is exactly \[ y(x)=W(a,D;x^{\alpha_{n+1}}), \] which is described there as the unique entire fundamental solution [2102.04347]. An application is also recorded to a nonlinear fractional PDE: \[ \frac{\partial u}{\partial t}+i\omega u = \beta\,{}^C\!D_x^a\!\bigl[e^{-ikx}u(x,t)\bigr]-a\,u(x,t), \qquad \omega,\beta,a,k\in\mathbb R, \] for which a separable periodic solution of period \(T=2\pi\) is given by \[ u(x,t)=e^{i\omega t}\,W(a,a,\nu;-ikx^a). \] The paper characterizes this solution as isochronous [2102.04347]. ## 3. Laplace transform and analytic relations A principal analytic relation established for \(\mathcal W^{(\bar\alpha,\bar\nu)}\) is its Laplace transform. For \(\Re s>0\), \[ \mathcal L\{\mathcal W^{(\bar\alpha,\bar\nu)}(\lambda x^{\alpha_{n+1}})\}(s) = \frac1s\sum_{k=0}^\infty A_k\Bigl(\frac{\lambda}{s^{\alpha_{n+1}}}\Bigr)^k, \] where the coefficients \(A_k\) are written explicitly as products of gamma quotients [2301.04640]. The transform becomes especially transparent when all \(\alpha_j=1\). In that case one recovers Delerue’s hyper-Bessel function, and the Laplace transform is expressed through the multi-index Mittag-Leffler function of Kiryakova–Luchko: \[ \mathcal L\{\mathcal W^{(\mathbf 1,\bar\nu)}(\lambda x)\}(s) = \prod_{j=1}^n\Gamma(1+a_j)\; \frac1s\; E_{(1,\dots,1),(a_2+1,\dots,a_{n+1}+1)}^{(n)}\!\Bigl(\frac{\lambda}{s}\Bigr). \] The summary emphasizes that this recovers, in particular cases, the well-known functional relation between the hyper-Bessel and multi-index Mittag-Leffler functions, and also the corresponding relation between the classical Wright and Mittag-Leffler functions [2301.04640]. A notable negative statement is equally part of the current record: no Mellin–Barnes or other contour integral representations are given in the 2023 paper beyond the Laplace integral. This circumscribes the analytic toolkit developed there. A plausible implication is that transform theory for \(\mathcal W^{(\bar\alpha,\bar\nu)}\) remains less developed than for the Fox–Wright class, where Mellin–Barnes methods are standard [2301.04640]. ## 4. Fractional recurrences and parameter differentiation In the three-parameter reduction \(\mathcal W_{\alpha,\beta,\nu}\), Droghei derives a recurrence relation involving Caputo derivatives \(D^\alpha\) and \(D^\beta\). The relation couples the shifted functions \(\mathcal W_{\alpha,\beta,\nu+\beta}\), \(\mathcal W_{\alpha,\beta,\nu}\), and \(\mathcal W_{\alpha,\beta,\nu-\beta}\), and is written as a three-term identity with weight factors in powers of \(x\) [2301.04640]. A distinguished specialization is obtained by setting \(\alpha=0\), \(\beta\to\alpha\), and \(\nu\to\beta-1\), which yields a new three-term differential recurrence for the two-parameter Mittag-Leffler function: \[ z^{\alpha}D^{\alpha}\!\Bigl[z^{\alpha+\beta-1}E_{\alpha,\alpha+\beta}(z^{\alpha})\Bigr] -2z^{\alpha+\beta-1}E_{\alpha,\beta}(z^{\alpha}) +z^{\beta-1}E_{\alpha,\beta-\alpha}(z^{\alpha}) =0. \] The source explicitly identifies this as new “to the best of our knowledge” [2301.04640]. The same work computes derivatives of \(\mathcal W_{\alpha,\beta,\nu}(z)\) with respect to the parameters \(\nu\), \(\beta\), and \(\alpha\). These are given as infinite power series whose coefficients involve quotients of gamma functions and the digamma function \[ \psi(w)=\frac{\Gamma'(w)}{\Gamma(w)}. \] In special cases, the parameter-derivative formulas reduce to known formulas for the Wright function due to Apelblat–Mainardi and for the Mittag-Leffler function due to Apelblat [2301.04640]. These formulas place \(\mathcal W_{\alpha,\beta,\nu}\) within the operational calculus of Wright- and Mittag-Leffler-type functions rather than treating it only as an isolated series. ## 5. Specializations and identifications A major theme of the subject is that the multi-index Wright function interpolates among several established special functions. The following reductions are explicitly recorded. | Specialization | Resulting function | Source | |---|---|---| | \(\alpha_j=1\) for all \(j\) | Delerue’s hyper-Bessel function of order \(d=n\) | [2301.04640] | | \(\alpha=1,\ \beta=\lambda,\ \nu=\mu\) | Classical Wright function \(W_{\lambda,\mu}\) | [2301.04640] | | \(\alpha=0,\ \beta\to\alpha,\ \nu\to\beta-1\) | Two-parameter Mittag-Leffler function \(E_{\alpha,\beta}\) | [2301.04640] | | \(\alpha=\beta=\nu\) | Garra–Polito type \(E_{1;\nu,1}(z)\) | [2301.04640] | | \(\alpha_j=\nu_j=1\) and \(\alpha_{n+1}=1\) | Laguerre–exponential \(e_n(z)\) | [2102.04347] | | \(\alpha_j=\alpha_{n+1}=1,\ \nu_1=\cdots=\nu_n=v\) | \(n\)-Mittag-Leffler function \(E_{n;v,1}(z)\) | [2102.04347] | | \(n=1,\ \alpha_1=1,\ \alpha_2=B,\ \nu_1=v\) | Classical Wright function \(W_{B,v}\) | [2102.04347] | | \(B=v=1\) in the previous line | Tricomi function \(C_0(x)\) | [2102.04347] | For the hyper-Bessel case, the 2023 summary states more specifically that \[ \mathcal W^{(\mathbf 1,\bar\nu)}(z) = \prod_{j=1}^n\Gamma(1+a_j)\,\mathcal J_{\mu_n}(-z), \] where \(\mathcal J_{\mu_n}\) is the order-\(n\) hyper-Bessel function [2301.04640]. For the classical Wright specialization, the reduction is written as \[ \mathcal W_{1,\lambda,\mu}(x^\lambda) = W_{\lambda,\mu}\!\bigl(x^\lambda/\lambda\bigr) = \sum_{k=0}^\infty \frac{(x^\lambda/\lambda)^k}{k!\,\Gamma(\lambda k+\mu)}. \] For the Mittag-Leffler specialization, \[ \mathcal W_{0,\alpha,\beta-1}(z)=E_{\alpha,\beta}(z). \] These identifications are significant because they place the multi-index Wright function inside the established hierarchy connecting hyper-Bessel, Wright, and Mittag-Leffler families [2301.04640]. ## 6. Terminology, Fox–Wright context, and asymptotic theory The phrase **multi-index Wright function** is terminologically ambiguous. In Droghei’s work it denotes the fractional-hyper-Bessel-derived family \(\mathcal W^{(\bar\alpha,\bar\nu)}\), whereas Paris uses the same phrase for the generalized Wright or Fox–Wright function \[ {}_p\Psi_q\!\bigl[(\alpha_r,a_r)_{r=1}^p;(\beta_r,b_r)_{r=1}^q;z\bigr] = \sum_{n=0}^\infty \frac{g(n)z^n}{n!}, \qquad g(n)=\frac{\prod_{r=1}^p\Gamma(\alpha_r n+a_r)}{\prod_{r=1}^q\Gamma(\beta_r n+b_r)}. \] This broader class is governed by the characteristic quantities \[ \kappa=1+\sum_{r=1}^q\beta_r-\sum_{r=1}^p\alpha_r,\qquad h=\frac{\prod_{r=1}^p\alpha_r^{\alpha_r}}{\prod_{r=1}^q\beta_r^{\beta_r}},\qquad \vartheta=\sum_{r=1}^q b_r-\sum_{r=1}^p a_r+\frac{q-p}{2}, \] together with \(\vartheta'=1-\vartheta\) [1708.04824]. Its convergence theory is classical: if \(\kappa>0\), the defining series converges absolutely for all finite \(z\); if \(\kappa=0\), it has finite radius \(|z|