Two-Parameter Mittag-Leffler Functions

• Two-parameter Mittag-Leffler functions are entire functions defined by a power series with complex parameters that generalize the exponential function and underpin models in fractional calculus.
• They exhibit rich analytic structures including uniform convergence, analytic continuation via Mellin-Barnes and Laplace representations, and allow termwise parameter differentiation.
• Robust numerical methods, such as Laplace transform inversion and Padé approximants, enable efficient evaluation in both scalar and matrix settings, benefiting applications in anomalous diffusion and operator theory.

The two-parameter Mittag-Leffler function, denoted $E_{\alpha,\beta}(z)$, is an entire function of a complex variable $z$ and complex parameters $\alpha$ and $\beta$. Its deep connections with fractional calculus, special function theory, and the modeling of anomalous physical processes have made it a central object in mathematics, applied analysis, and computational science. The function exhibits rich analytic structure—including uniform convergence, intricate parameter dependence, and a spectrum of representation and approximation techniques—enabling its use in both theoretical and computational contexts.

1. Definition, Series Representation, and Basic Properties

The two-parameter Mittag-Leffler function is defined by the absolutely convergent power series

$$E_{\alpha,\beta}(z) = \sum_{k=0}^\infty \frac{z^k}{\Gamma(\alpha k + \beta)},$$

where $\alpha>0$, $\beta\in\mathbb{C}$, and $z\in\mathbb{C}$ (Rogosin et al., 29 Jul 2024, Lavault, 2017, Garrappa et al., 2 Oct 2024). The function is entire in $z$ for all fixed parameters with $\operatorname{Re}\, \alpha > 0$, and its order as an entire function is $1/\operatorname{Re}\alpha$ (Lavault, 2017, Garrappa et al., 2 Oct 2024). The factorial growth of the Gamma function in the denominator ensures convergence for all $z$.

Special cases elucidate its generality:

• $E_{1,1}(z)=e^z$,
• $E_{2,1}(z^2)=\cosh z$, $E_{2,1}(-z^2)=\cos z$,
• $$E_{1/2,1}(\pm z^{1/2})=e^z\operatorname{erfc}(\mp z^{1/2})$$ (Garrappa, 2015).

For $\operatorname{Re}(z)\to -\infty$, the function decays rapidly, and for large $|z|$ its behavior can be characterized by asymptotic expansions and integral representations.

2. Analytic Continuation and Integral Representations

Apart from the series definition, several integral representations provide analytic continuation and access to various parameter ranges. Two principal families include:

• Mellin–Barnes type: For $\operatorname{Re}\, \alpha > 0$, $\operatorname{Re}\, \beta > 0$, $|\arg z|<\pi$,

$$E_{\alpha,\beta}(z) = \frac{1}{2\pi i} \int_{C} \Gamma(s) \Gamma(1-s) \Gamma(\beta-\alpha s) (-z)^{-s}\, ds,$$

with $C$ a vertical line $0<\operatorname{Re}s<1$ (Rogosin et al., 29 Jul 2024).

• Contour integrals of "Wiman" or "Hankel" type: Multiple real and complex forms arise, including representations adapted for numerical inversion of Laplace transforms and for various (sub)domains in $z$ (Saenko, 2020). For example, in the symmetric case, representation "B" (single improper real integral) is valid when $\operatorname{Re}\mu < 1+1/\rho$:

$$E_{\rho,\mu}(z) = \int_{0}^{\infty} K_{\rho,\mu}(r,\delta_\rho;z)\, dr,$$

where $K$ is an explicit real kernel (Saenko, 2020).

• Laplace representation (for $0<\alpha\le 1$, $\beta\ge\alpha$):

$$\Gamma(\beta) E_{\alpha,\beta}(-x) = \mathbb{E}[e^{-x\, M_{\alpha,\beta}}],$$

giving probabilistic and monotonicity properties (Garrappa et al., 2 Oct 2024).

These representations provide analytic continuation, explicit means to compute or analyze $E_{\alpha,\beta}(z)$ in various regions, and route to parameter derivatives.

3. Parameter Differentiation and Uniform Convergence

The dependence of $E_{\alpha,\beta}(z)$ on its parameters is highly regular. For $(\alpha, \beta)$ in open subsets of $\mathbb{C}^2$, the function is smooth in both parameters and $z$. Termwise differentiation with respect to $\alpha$ and $\beta$ is justified by uniform convergence of the differentiated series on compact sets (Rogosin et al., 29 Jul 2024), leading to: $$\frac{\partial}{\partial\alpha} E_{\alpha,\beta}(z) = -\sum_{k=1}^\infty k\,\psi(\alpha k + \beta) \,\frac{z^k}{\Gamma(\alpha k + \beta)},$$

$$\frac{\partial}{\partial\beta} E_{\alpha,\beta}(z) = -\sum_{k=0}^\infty \psi(\alpha k + \beta) \frac{z^k}{\Gamma(\alpha k + \beta)},$$

where $\psi(x) = \Gamma'(x)/\Gamma(x)$ is the digamma function. These differentiated series are absolutely and uniformly convergent for $z$ in compact sets of $\mathbb{C}$ and suitable parameter regions.

Alternatively, Mellin–Barnes representations yield derivatives under the integral sign, crucial for deriving asymptotics and establishing analytical properties beyond the domain of the pure series (Rogosin et al., 29 Jul 2024).

4. Inequalities, Monotonicity, and Log-Convexity

The comparative size of $|E_{\alpha,\beta}(z)|$ and $E_{\alpha,\beta}(\Re z)$ is characterized by sharp inequalities and depends delicately on $(\alpha,\beta)$:

• The inequality $|E_{\alpha,\beta}(z)| \le E_{\alpha,\beta}(\Re z)$ for all $z$ holds if and only if $E_{\alpha,\beta}(-x)$ is completely monotone on $(0,\infty)$, i.e., when $0<\alpha\le 1$, $\beta\ge\alpha$ (Garrappa et al., 2 Oct 2024). In this regime, $E_{\alpha,\beta}$ admits a Laplace transform representation.
• The reverse inequality, $|E_{\alpha,\beta}(z)| \ge E_{\alpha,\beta}(\Re z)$, holds for
  • $\alpha=1$, $\beta\le 1$,
  • all $z$ if $\alpha\ge 2$ and $\beta$ within an explicit range ensuring all zeros are real and negative,
  • and conjecturally for $1<\alpha<2$, $\beta\in[\alpha-1,\alpha]$ when $1/E_{\alpha,\beta}(x)$ is completely monotone.
• The function $E_{\alpha,\beta}(x)$ is log-convex on $(0,\infty)$ if and only if $\alpha\le 1$ and $\beta \ge h(\alpha)$, where $h(\alpha)$ is defined by an implicit transcendental condition involving the Gamma function; it is log-concave precisely for $\alpha\ge 1$ and $\beta\le h(\alpha)$ (Garrappa et al., 2 Oct 2024).

These sharp domains are significant for applications in fractional dynamics and for establishing subordination properties.

5. Asymptotics, Oscillatory Behavior, and Zeros

The large-$|z|$ asymptotic expansion, valid for $|z|\to\infty$ in sectors away from the positive real axis, is given by (Sarumi et al., 2019, Lavault, 2017): $$E_{\alpha,\beta}(z) = -\sum_{k=1}^n \frac{z^{-k}\Gamma(\beta-\alpha k)}{\Gamma(\beta)} + O(|z|^{-n-1}),$$ with corresponding shifts for special parameter relations.

For $\alpha>1$, $E_{\alpha,\beta}(-t)$ may cease to be monotone, acquiring oscillatory decay and finitely many real zeros. The "derooting" decomposition expresses the function as a sum of a low-degree polynomial (carrying all real zeros) and a shifted term that has no real zeros and decays monotonically (Honain et al., 2023): $$E_{\alpha,\beta}(-t) = (-t)^r E_{\alpha, \beta + \alpha r}(-t) + P^{r-1}_{\alpha,\beta}(-t),$$ where $$P^{r-1}_{\alpha,\beta}(t) = \sum_{k=0}^{r-1} t^k / \Gamma(\alpha k + \beta)$$ and $r$ is chosen to shift parameters into a non-oscillatory regime.

Phase diagrams and analysis of the location of zeros and oscillations have been developed in detail for $1<\alpha<2$ (Honain et al., 2023).

6. Numerical Evaluation and Rational Approximation

The numerical evaluation of $E_{\alpha,\beta}(z)$ is fundamentally challenging for large $|z|$, highly oscillatory regimes, or for matrix arguments in operator-theoretic applications.

• Direct Taylor Expansion: Highly effective for small $|z|$ due to rapid convergence of the defining series.
• Laplace Transform Inversion (Optimal Parabolic Contour, OPC): For general complex $z$, the OPC method inverts the Laplace transform along an optimally chosen parabolic contour, guaranteeing machine-precision accuracy with modest computational effort. This is automatic, robust across the full parameter range, and also extends to the three-parameter Prabhakar function (Garrappa, 2015).
• Global Padé and Rational Approximants: For $0<\alpha\le1$, $\beta\ge\alpha$, accurate global Padé approximants are constructed by matching Maclaurin expansion and asymptotic conditions (e.g., fourth-order approximants $R^{5,4}$, $R^{6,3}$, $R^{7,2}$). These approximants, via partial fractions, enable fast and accurate evaluation even for large matrices (Honain et al., 2023, Sarumi et al., 2019).
• Derooting plus Padé: In non-monotone regimes $\alpha>1$, the derooting decomposition is coupled with rational approximants to capture zeros and oscillatory behavior robustly for scalar and matrix arguments (Honain et al., 2023).
• Matrix Argument Evaluation: For $A\in\mathbb{C}^{n\times n}$, scalar contour formulas carry over via the Cauchy integral, or via Schur-Parlett reduction plus optimized scalar routines for block-triangular atomic blocks. Recent developments combine Taylor-series, Schur decomposition, and trapezoidal quadrature on optimal contours to achieve derivative-free accuracy in IEEE double precision (Cardoso, 2023).

Empirical results show that these rational approximants yield uniform errors on the order of $10^{-6}$–$10^{-8}$ for $t\in[0,20]$ in non-oscillatory regimes, and controlled accuracy for oscillatory problems, with speedups of at least $80\times$ over classic methods for large-matrix settings (Honain et al., 2023, Sarumi et al., 2019).

7. Extensions and Applications

The analytic and algorithmic techniques for the two-parameter Mittag-Leffler function extend naturally to broader classes:

• Multi-variable and Generalized Mittag-Leffler Functions: Double series and contour representations accommodate higher-dimensional generalizations (Lavault, 2017).
• Prabhakar (Three-Parameter) and Four-Parameter Wright Functions: The analytic framework—especially uniform convergence arguments for termwise parameter differentiation and contour integral representations—extends to these cases (Rogosin et al., 29 Jul 2024).
• Fractional Differential Equations and Operator Theory: $E_{\alpha,\beta}$ functions are kernels of solutions to fractional diffusion and wave equations, and tools for spectral and numerical integration (Honain et al., 2023, Sarumi et al., 2019, Cardoso, 2023).
• Construction of Special Functions: Differentiation in parameters aids the systematic derivation of novel families, as well as in the study of asymptotics, convexity, and monotonicity properties relevant in complex analysis and probability (Rogosin et al., 29 Jul 2024, Garrappa et al., 2 Oct 2024).

Further, the function’s role as a model for memory and nonlocal effects makes it a pillar for fractional calculus, viscoelasticity, anomalous diffusion, and mathematical physics.

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References:

• (Rogosin et al., 29 Jul 2024) On differentiation with respect to parameters of the functions of the Mittag-Leffler type
• (Lavault, 2017) Integral representations and asymptotic behaviour of a Mittag-Leffler type function of two variables
• (Honain et al., 2023) Rational Approximations for Oscillatory Two-Parameter Mittag-Leffler Function
• (Sarumi et al., 2019) Highly Accurate Global Padé Approximations of Generalized Mittag-Leffler Function and its Inverse
• (Cardoso, 2023) Computing the Mittag-Leffler Function of a Matrix Argument
• (Saenko, 2020) Two forms of the integral representations of the Mittag-Leffler function
• (Garrappa, 2015) Numerical evaluation of two and three parameter Mittag-Leffler functions
• (Garrappa et al., 2 Oct 2024) On some inequalities for the two-parameter Mittag-Leffler function in the complex plane