On $${\varvec{q}}$$ q -congruences related to $${\varvec{H}}_{\varvec{n}}\left( {\varvec{x;q}}\right) $$ H n x ; q and $${\varvec{M}}_{{\varvec{n}}}\left( {\varvec{x;q}}\right) $$ M n x ; q

Sibel Koparal (), Neşe Ömür () and Laid Elkhiri ()
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Sibel Koparal: University of Bursa Uludağ
Neşe Ömür: University of Kocaeli
Laid Elkhiri: University of Tiaret

Indian Journal of Pure and Applied Mathematics, 2025, vol. 56, issue 2, 778-790

Abstract: Abstract In this paper, we examine $$\sum \limits _{k=m}^{p-1}q^{k} {k \brack m} _{q}H_{k}\left( x;q\right) \pmod {\left[ p\right] _{q}}$$ ∑ k = m p - 1 q k k m q H k x ; q ( mod p q ) and $$M_{p-1}\left( x;q\right) \pmod {\left[ p\right] _{q}},$$ M p - 1 x ; q ( mod p q ) , where for real number x, $$H_{n}\left( x;q\right) =\sum \limits _{k=1}^{n}\frac{x^{k}}{\left[ k\right] _{q}}$$ H n x ; q = ∑ k = 1 n x k k q and $$M_{n}\left( x;q\right) =\sum \limits _{k=1}^{n}k\frac{x^{k}}{\left[ k\right] _{q}}.$$ M n x ; q = ∑ k = 1 n k x k k q . For example, for an odd prime number p and $$m\in \left\{ 0,1,...,p-2\right\} ,$$ m ∈ 0 , 1 , . . . , p - 2 , $$\begin{aligned}{} & {} \left[ m+1\right] _{q}\sum \limits _{k=m}^{p-1}q^{k} {k \brack m} _{q}H_{k}\left( x;q\right) \\{} & {} \quad \equiv \left( -1\right) ^{m}q^{-\left( {\begin{array}{c}m\\ 2\end{array}}\right) }\left( 1-\left( x;q\right) _{p}\right) -\frac{q^{m}}{\left( xq;q\right) _{m}}x^{m+1}\left[ p\right] _{x} \pmod {\left[ p\right] _{q}}. \end{aligned}$$ m + 1 q ∑ k = m p - 1 q k k m q H k x ; q ≡ - 1 m q - m 2 1 - x ; q p - q m x q ; q m x m + 1 p x ( mod p q ) .

Keywords: Congruence; $$q-$$ q - analog number; Harmonic number; 11A07; 11B65 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s13226-023-00520-0

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