Multi-index Wright Function: Fractional Hyper-Bessel

• The multi-index Wright function is a multiple-parameter entire function defined via gamma ratios and fractional hyper-Bessel operators, crucial for solving eigenvalue problems in fractional calculus.
• It connects with classical functions like the Wright, Mittag-Leffler, and Fox–Wright functions through systematic parameter reductions and specific recurrence relations.
• Its analysis via Laplace transforms and fractional recurrences highlights its theoretical importance and practical applications in modeling fractional differential equations.

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 (Droghei, 2023). 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 (Paris, 2017).

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 $\bar\alpha=(\alpha_1,\dots,\alpha_{n+1})$0, $\bar\alpha=(\alpha_1,\dots,\alpha_{n+1})$1, and the terminal factor involving $\bar\alpha=(\alpha_1,\dots,\alpha_{n+1})$2 (Droghei, 2023).

A central structural fact is that $\bar\alpha=(\alpha_1,\dots,\alpha_{n+1})$3 is entire. The summary of Droghei’s results states that, since $\bar\alpha=(\alpha_1,\dots,\alpha_{n+1})$4, the ratio test yields convergence for all $\bar\alpha=(\alpha_1,\dots,\alpha_{n+1})$5. In the related formulation $\bar\alpha=(\alpha_1,\dots,\alpha_{n+1})$6 from the 2021 paper, entireness is again established by the D’Alembert criterion together with Wendel’s asymptotic formula, under $\bar\alpha=(\alpha_1,\dots,\alpha_{n+1})$7 and $\bar\alpha=(\alpha_1,\dots,\alpha_{n+1})$8 (Droghei, 2021).

The three-parameter reduction

$\bar\alpha=(\alpha_1,\dots,\alpha_{n+1})$9

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 (Droghei, 2023).

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 (Droghei, 2021). In the notation of the 2021 paper, the operator is

$\bar\nu=(\nu_1,\dots,\nu_n)$0

where $\bar\nu=(\nu_1,\dots,\nu_n)$1 denotes the Caputo derivative.

Its action on monomials is computed explicitly: $\bar\nu=(\nu_1,\dots,\nu_n)$2 Summing termwise in the defining series yields the operator identity

$\bar\nu=(\nu_1,\dots,\nu_n)$3

Hence, whenever $\bar\nu=(\nu_1,\dots,\nu_n)$4, the function becomes an eigenfunction of $\bar\nu=(\nu_1,\dots,\nu_n)$5 (Droghei, 2021).

The same paper presents the associated fractional ODE

$\bar\nu=(\nu_1,\dots,\nu_n)$6

with initial condition $\bar\nu=(\nu_1,\dots,\nu_n)$7 and all lower Caputo derivatives at $\bar\nu=(\nu_1,\dots,\nu_n)$8 equal to zero. The power-series solution is exactly

$\bar\nu=(\nu_1,\dots,\nu_n)$9

which is described there as the unique entire fundamental solution (Droghei, 2021).

An application is also recorded to a nonlinear fractional PDE: ${}_p\Psi_q(z)$0 for which a separable periodic solution of period ${}_p\Psi_q(z)$1 is given by

${}_p\Psi_q(z)$2

The paper characterizes this solution as isochronous (Droghei, 2021).

3. Laplace transform and analytic relations

A principal analytic relation established for ${}_p\Psi_q(z)$3 is its Laplace transform. For ${}_p\Psi_q(z)$4,

${}_p\Psi_q(z)$5

where the coefficients ${}_p\Psi_q(z)$6 are written explicitly as products of gamma quotients (Droghei, 2023).

The transform becomes especially transparent when all ${}_p\Psi_q(z)$7. 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: ${}_p\Psi_q(z)$8 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 (Droghei, 2023).

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 ${}_p\Psi_q(z)$9 remains less developed than for the Fox–Wright class, where Mellin–Barnes methods are standard (Droghei, 2023).

4. Fractional recurrences and parameter differentiation

In the three-parameter reduction $$\bar\alpha=(\alpha_1,\dots,\alpha_{n+1}),\qquad \bar\nu=(\nu_1,\dots,\nu_n),$$0, Droghei derives a recurrence relation involving Caputo derivatives $$\bar\alpha=(\alpha_1,\dots,\alpha_{n+1}),\qquad \bar\nu=(\nu_1,\dots,\nu_n),$$1 and $$\bar\alpha=(\alpha_1,\dots,\alpha_{n+1}),\qquad \bar\nu=(\nu_1,\dots,\nu_n),$$2. The relation couples the shifted functions $$\bar\alpha=(\alpha_1,\dots,\alpha_{n+1}),\qquad \bar\nu=(\nu_1,\dots,\nu_n),$$3, $$\bar\alpha=(\alpha_1,\dots,\alpha_{n+1}),\qquad \bar\nu=(\nu_1,\dots,\nu_n),$$4, and $$\bar\alpha=(\alpha_1,\dots,\alpha_{n+1}),\qquad \bar\nu=(\nu_1,\dots,\nu_n),$$5, and is written as a three-term identity with weight factors in powers of $$\bar\alpha=(\alpha_1,\dots,\alpha_{n+1}),\qquad \bar\nu=(\nu_1,\dots,\nu_n),$$6 (Droghei, 2023).

A distinguished specialization is obtained by setting $$\bar\alpha=(\alpha_1,\dots,\alpha_{n+1}),\qquad \bar\nu=(\nu_1,\dots,\nu_n),$$7, $$\bar\alpha=(\alpha_1,\dots,\alpha_{n+1}),\qquad \bar\nu=(\nu_1,\dots,\nu_n),$$8, and $$\bar\alpha=(\alpha_1,\dots,\alpha_{n+1}),\qquad \bar\nu=(\nu_1,\dots,\nu_n),$$9, which yields a new three-term differential recurrence for the two-parameter Mittag-Leffler function: $\alpha_0=\nu_0=0$0 The source explicitly identifies this as new “to the best of our knowledge” (Droghei, 2023).

The same work computes derivatives of $\alpha_0=\nu_0=0$1 with respect to the parameters $\alpha_0=\nu_0=0$2, $\alpha_0=\nu_0=0$3, and $\alpha_0=\nu_0=0$4. These are given as infinite power series whose coefficients involve quotients of gamma functions and the digamma function

$\alpha_0=\nu_0=0$5

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 (Droghei, 2023).

These formulas place $\alpha_0=\nu_0=0$6 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_0=\nu_0=0$7 for all $\alpha_0=\nu_0=0$8	Delerue’s hyper-Bessel function of order $\alpha_0=\nu_0=0$9	(Droghei, 2023)
$$a_j=1+\sum_{m=1}^j(\nu_{m-1}-\alpha_m),\qquad b_j=1+\sum_{m=1}^j(\nu_{m-1}-\alpha_{m-1}),$$0	Classical Wright function $$a_j=1+\sum_{m=1}^j(\nu_{m-1}-\alpha_m),\qquad b_j=1+\sum_{m=1}^j(\nu_{m-1}-\alpha_{m-1}),$$1	(Droghei, 2023)
$$a_j=1+\sum_{m=1}^j(\nu_{m-1}-\alpha_m),\qquad b_j=1+\sum_{m=1}^j(\nu_{m-1}-\alpha_{m-1}),$$2	Two-parameter Mittag-Leffler function $$a_j=1+\sum_{m=1}^j(\nu_{m-1}-\alpha_m),\qquad b_j=1+\sum_{m=1}^j(\nu_{m-1}-\alpha_{m-1}),$$3	(Droghei, 2023)
$$a_j=1+\sum_{m=1}^j(\nu_{m-1}-\alpha_m),\qquad b_j=1+\sum_{m=1}^j(\nu_{m-1}-\alpha_{m-1}),$$4	Garra–Polito type $$a_j=1+\sum_{m=1}^j(\nu_{m-1}-\alpha_m),\qquad b_j=1+\sum_{m=1}^j(\nu_{m-1}-\alpha_{m-1}),$$5	(Droghei, 2023)
$$a_j=1+\sum_{m=1}^j(\nu_{m-1}-\alpha_m),\qquad b_j=1+\sum_{m=1}^j(\nu_{m-1}-\alpha_{m-1}),$$6 and $$a_j=1+\sum_{m=1}^j(\nu_{m-1}-\alpha_m),\qquad b_j=1+\sum_{m=1}^j(\nu_{m-1}-\alpha_{m-1}),$$7	Laguerre–exponential $$a_j=1+\sum_{m=1}^j(\nu_{m-1}-\alpha_m),\qquad b_j=1+\sum_{m=1}^j(\nu_{m-1}-\alpha_{m-1}),$$8	(Droghei, 2021)
$$a_j=1+\sum_{m=1}^j(\nu_{m-1}-\alpha_m),\qquad b_j=1+\sum_{m=1}^j(\nu_{m-1}-\alpha_{m-1}),$$9	$j=1,\dots,n+1$0-Mittag-Leffler function $j=1,\dots,n+1$1	(Droghei, 2021)
$j=1,\dots,n+1$2	Classical Wright function $j=1,\dots,n+1$3	(Droghei, 2021)
$j=1,\dots,n+1$4 in the previous line	Tricomi function $j=1,\dots,n+1$5	(Droghei, 2021)

For the hyper-Bessel case, the 2023 summary states more specifically that

$j=1,\dots,n+1$6

where $j=1,\dots,n+1$7 is the order-$j=1,\dots,n+1$8 hyper-Bessel function (Droghei, 2023). For the classical Wright specialization, the reduction is written as

$j=1,\dots,n+1$9

For the Mittag-Leffler specialization,

$a_j=b_j-\alpha_j$0

These identifications are significant because they place the multi-index Wright function inside the established hierarchy connecting hyper-Bessel, Wright, and Mittag-Leffler families (Droghei, 2023).

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 $a_j=b_j-\alpha_j$1, whereas Paris uses the same phrase for the generalized Wright or Fox–Wright function

$a_j=b_j-\alpha_j$2

This broader class is governed by the characteristic quantities

$a_j=b_j-\alpha_j$3

together with $a_j=b_j-\alpha_j$4 (Paris, 2017).

Its convergence theory is classical: if $a_j=b_j-\alpha_j$5, the defining series converges absolutely for all finite $a_j=b_j-\alpha_j$6; if $a_j=b_j-\alpha_j$7, it has finite radius $a_j=b_j-\alpha_j$8; if $a_j=b_j-\alpha_j$9, it diverges for every $\mathcal W^{(\bar\alpha,\bar\nu)}(z)$0 (Paris, 2017). Its large-$\mathcal W^{(\bar\alpha,\bar\nu)}(z)$1 behavior is described through an exponential expansion $\mathcal W^{(\bar\alpha,\bar\nu)}(z)$2, an algebraic expansion $\mathcal W^{(\bar\alpha,\bar\nu)}(z)$3, and Stokes switching. For $\mathcal W^{(\bar\alpha,\bar\nu)}(z)$4, the rays $\mathcal W^{(\bar\alpha,\bar\nu)}(z)$5 are Stokes lines, and the subdominant exponential contribution undergoes error-function smoothing across a Berry region of thickness $\mathcal W^{(\bar\alpha,\bar\nu)}(z)$6 (Paris, 2017).

Paris’s numerical study shows that these exponentially small terms are not merely formal corrections: in the examples treated there, including Mittag-Leffler and other $\mathcal W^{(\bar\alpha,\bar\nu)}(z)$7 cases, accurate evaluation requires the subdominant exponentials and their Stokes multipliers (Paris, 2017). A plausible implication for the narrower function $\mathcal W^{(\bar\alpha,\bar\nu)}(z)$8 is that an analogous asymptotic theory would be valuable, but the 2023 study does not yet provide Wright–Braaksma-type asymptotic expansions.

Taken together, the current literature presents the multi-index Wright function in two complementary senses: as a concrete special-function family tied to fractional hyper-Bessel eigenproblems, and as part of the larger Wright/Fox–Wright ecosystem in which convergence sectors, exponential asymptotics, and Stokes phenomena are already highly developed.