| Dedicated to Professor Constantin Fetecau on the occasion of his 80th anniversary
Mircea Merca†
Abstract: We show that explicit expansions of the Chebyshev polynomials of the first kind naturally induce several invertible triangular transformations involving classical binomial arrays and closely related ballot-number arrays. Using a unified coefficient-extraction method, we derive direct and inverse formulas relating the coefficients (2n
n−k) and the ballot-number differences (2n
n+k) − (2n
n+k+1) to the sequences (n
k)2n−k and (n
k)4n−k. The resulting identities define explicit mutually inverse triangular transformations. The closed forms appearing on their right-hand sides admit natural combinatorial interpretations in terms of disjoint subsets and block decompositions. We also introduce a signed refinement of the classical enumeration of pairs of disjoint subsets, which yields an additional Chebyshev–binomial transformation and its explicit inverse.
Keywords: Chebyshev polynomials, symmetric Laurent polynomials, binomial transformations, combinatorial interpretations.
MSC: 055A19, 11B65, 33C45, 05A15.
DOI 10.56082/annalsarscimath.2026.4.149
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*Accepted for publication on July 17, 2026
†mircea.merca@upb.ro, Department of Mathematical Models and Methods, Fundamental Sciences Applied in Engineering Research Center, National University of Science and Technology Politehnica Bucharest, 060042 Bucharest, Romania and Academy of Romanian Scientists, 050044 Bucharest, Romania |
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PUBLISHED in
Annals Academy of Romanian Scientists Series on Mathematics and Its Application,
Volume 18 no 4, 2026
  
ISSN ONLINE 2066 – 6594
ISSN PRINT 2066 – 5997
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