Structural Recurrence Identities for the Seven-Parameter Mittag–Leffler Function

Résumé

The seven-parameter Mittag–Leffler function, a recent extension of classical fractional kernels,
offers a wide range of structural flexibility but still lacks a coherent set of identities linking neighbouring
parameter configurations. This gap limits both theoretical analysis and the efficiency of numerical schemes in
models where slight parameter shifts occur naturally. In this work, we derive four recurrence relations that
describe how the function transforms under shifts in its order parameters and under successive differentiations.
Each identity follows directly from the internal organisation of the defining series and the interplay between
Pochhammer and Gamma structures.
The resulting formulas provide a compact mechanism for reconstructing kernels without recomputing full
series expansions, and they remain stable under numerical evaluation. Their utility is demonstrated through
examples relevant to viscoelasticity, anomalous transport, electromagnetic dispersion and other systems governed by nonlocal laws. These relations form a foundation for streamlined computation and open a path
toward deeper structural theory for multi-parameter Mittag–Leffler families.