On q -congruences related to Hn(x;q) and Mn(x;q)

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Koparal S., ÖMÜR N., Elkhiri L.

Indian Journal of Pure and Applied Mathematics, vol.56, no.2, pp.778-790, 2025 (SCI-Expanded, Scopus)

• Publication Type: Article / Article
• Volume: 56 Issue: 2
• Publication Date: 2025
• Doi Number: 10.1007/s13226-023-00520-0
• Journal Name: Indian Journal of Pure and Applied Mathematics
• Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, INSPEC, zbMATH
• Page Numbers: pp.778-790
• Keywords: Congruence, Harmonic number, q-analog number
• Kocaeli University Affiliated: Yes

Abstract

In this paper, we examine ∑k=mp-1qk[km]qHk(x;q)(mod[p]q) and Mp-1(x;q)(mod[p]q), where for real number x, Hn(x;q)=∑k=1nxk[k]q and Mn(x;q)=∑k=1nkxk[k]q. For example, for an odd prime number p and m∈ { 0 , 1 ,.. , p- 2 } , [m+1]q∑k=mp-1qk[km]qHk(x;q)≡(-1)mq-(m2)(1-(x;q)p)-qm(xq;q)mxm+1[p]x(mod[p]q).