From special-function recurrences to arithmetic continued fractions

Nurdaulet Shynarbek; Alibek Orynbassar; Muhammad Ateeq Tahir; Nurdaulet Shynarbek; Alibek Orynbassar; Muhammad Ateeq Tahir

doi:10.3934/era.2026235

Electronic Research Archive

2026, Volume 34, Issue 8: 5286-5302. doi: 10.3934/era.2026235

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From special-function recurrences to arithmetic continued fractions

1.
Department of Pedagogy of Natural Sciences, SDU University, Kaskelen, Kazakhstan
2.
School of Digital Technologies, Narxoz University, Almaty, Kazakhstan

Received: 13 February 2026 Revised: 28 May 2026 Accepted: 05 June 2026 Published: 22 June 2026

Many special functions are defined by differential equations, but useful arithmetic continued fractions often come from their discrete recurrences. This paper studied Riccati-type recurrences and their continued-fraction expansions in the setting of machine-discovered patterns from the enumerated signed-continued-fraction Massey approve algorithm (ESMA). The main contribution was a block method for Riccati recurrences, which was used to prove identities for signed interlaced continued fractions with affine partial denominators. We showed that several ESMA-type conjectures can be reduced to finite matrix identities in the block index. Three representative identities were proved in detail. They involved modified Bessel functions, half-integer Bessel functions, and ratios of integer-order Bessel functions. We also gave positivity checks and computable tail estimates. These estimates justified the passage to the infinite limit and gave explicit truncation bounds. The results showed that the representative formulas studied here were not isolated numerical coincidences.
- Riccati difference equation,
- continued fractions,
- Möbius iteration,
- stability,
- truncation error,
- special functions
Citation: Nurdaulet Shynarbek, Alibek Orynbassar, Muhammad Ateeq Tahir. From special-function recurrences to arithmetic continued fractions[J]. Electronic Research Archive, 2026, 34(8): 5286-5302. doi: 10.3934/era.2026235

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Abstract

Many special functions are defined by differential equations, but useful arithmetic continued fractions often come from their discrete recurrences. This paper studied Riccati-type recurrences and their continued-fraction expansions in the setting of machine-discovered patterns from the enumerated signed-continued-fraction Massey approve algorithm (ESMA). The main contribution was a block method for Riccati recurrences, which was used to prove identities for signed interlaced continued fractions with affine partial denominators. We showed that several ESMA-type conjectures can be reduced to finite matrix identities in the block index. Three representative identities were proved in detail. They involved modified Bessel functions, half-integer Bessel functions, and ratios of integer-order Bessel functions. We also gave positivity checks and computable tail estimates. These estimates justified the passage to the infinite limit and gave explicit truncation bounds. The results showed that the representative formulas studied here were not isolated numerical coincidences.

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