Generalized n-th Kind Extended q-Difference Operator: Unified Theory and Applications

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DOI: https://doi.org/10.28924/2291-8639-24-2026-103
Published: Apr 7, 2026
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J. Kathiravan, Khadar Babu Shaik, Generalized n-th Kind Extended q-Difference Operator: Unified Theory and Applications, Int. J. Anal. Appl., 24 (2026), 103.

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J. Kathiravan, Khadar Babu Shaik

Abstract

This work defines the extended \(q\)-difference operator of \(n\)-\(th\) kind \(\Delta_{q(\ell_{1},\ell_{2},\ell_{3},\cdots,\ell_{n})}\) and presents the discrete version of the Leibniz theorem according to \(\Delta_{q(\ell_{1},\ell_{2},\ell_{3},\cdots,\ell_{n})}\), for \(q\) and \(\ell_{i}^{'s}\) are positive reals. Derived some interesting results on the relation between the extended \(q\)-polynomial factorial of the first and \(n\)-\(th\) kind. The reciprocal of the extended \(q\)-difference operator of \(n\)-\(th\) kind, \(\Delta^{-1}_{q(\ell_{1},\ell_{2},\ell_{3},\cdots,\ell_{n})}\), is also described. The derivation of the formula for the sum of higher powers of arithmetic progressions is extended through numerical methods, supported by illustrative examples that emphasize the principal results.

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