Congruences related to dual sequences and Catalan numbers
During the study of dual sequences, Z.-W. Sun introduced the polynomials Dn(x,y)=∑k=0nnkxkykandSn(x,y)=∑k=0nnkxk−1−xkyk.Many related congruences were also established and conjectured. Here we generalize some of them by determining ∑k=0p−1Dk(x1,y1)Dk(x2,y2)(modp)and∑k=0p−1Sk(x1,y1)Sk(x2,y2)(modp)for...
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| Published in: | European journal of combinatorics Vol. 101; p. 103458 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Elsevier Ltd
01-03-2022
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| Online Access: | Get full text |
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| Summary: | During the study of dual sequences, Z.-W. Sun introduced the polynomials Dn(x,y)=∑k=0nnkxkykandSn(x,y)=∑k=0nnkxk−1−xkyk.Many related congruences were also established and conjectured. Here we generalize some of them by determining ∑k=0p−1Dk(x1,y1)Dk(x2,y2)(modp)and∑k=0p−1Sk(x1,y1)Sk(x2,y2)(modp)for odd primes p and p-adic integers xi,yi with i∈{1,2}. In addition, we also characterize ∑n=0p−1∑k=0nnkCkak2(modp),where Ck denotes the kth Catalan number, a∈Z∖{0} with gcd(a,p)=1. These results confirm and generalize some of Z.-W. Sun’s conjectures. |
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| ISSN: | 0195-6698 1095-9971 |
| DOI: | 10.1016/j.ejc.2021.103458 |