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arXiv · 2610.39023

An involution for recurrences of Tricomi and other special functions

John T. Conway
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Abstract

Recently two new integrated recurrences were given for the Kummer function which together with the six recurrences already given in standard literature gave a total of eight recurrences, each one linking the baseline function to one of it's eight contiguous functions. Two new recurrences for the Tricomi function are given here which complete the set of eight recurrences for the Tricomi function linking it to it's contiguous functions. The involution presented previously is examined in detail for the recurrences of Cylinder, Associated Legendre and Kummer/Tricomi functions. This involution converts any given recurrence to a recurrence with the parameter shifts reversed, which creates conjugate pairs. One pair for Cylinder functions, two pairs for Associated Legendre functions and 4 pairs each for Kummer and Tricomi functions. Uniqueness is proven using Wronskians for the integrated recurrence relations, such that contiguous functions are single occupancy slots with only one integrated recurrence possible for each slot and for each independent solution of the ODE. For Cylinder and Associated Legendre functions this is simple, as the recurrences are identical for each independent solution. However, for Kummer and Tricomi functions the recurrences are different. This difficulty is bypassed using Gamma functions to scale the Tricomi recurrences to match the Kummer recurrences for each conjugate pair. It is not possible to match globally. This allows uniqueness to be proven using Wronskians on a pair by pair basis. Hence the involution system behaves like a Pauli Exclusion principle for contiguous functions, with only one recurrence possible for each solution of the ODE.

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John T. Conway. 2026-10-28. An involution for recurrences of Tricomi and other special functions. https://arxiv.org/abs/2610.39023

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