𝑞-Bernoulli and Eulerian numbers

Carlitz, L.

doi:10.1090/s0002-9947-1954-0060538-2

Cited by 234 publications

(342 citation statements)

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“…Our second goal is to provide combinatorial interpretations for those two sequences of polynomials (A * n (t, q)), ((−1) n A * 2n+1 (−1, q)), compatible with what is already known for A n (t) and T 2n+1 . As we now see, the polynomials A * n (t, q), we have found, are slight variations of the classical qanalogs A n (t, q) of the Eulerian polynomials introduced by Carlitz [Ca54]. The latter polynomials may be defined by the identity (1.6)…”

Section: Since Euler Both Eulerian Polynomialssupporting

confidence: 52%

Doubloons and New Q-Tangent Numbers

Foata¹,

Han²

2009

The Quarterly Journal of Mathematics

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ABSTRACT. We introduce new q-tangent numbers based on the Carlitz q-analog of the Eulerian polynomial and the so-called doubloon combinatorial set-up. Those new q-tangent numbers are polynomials with positive integral coefficients. They are divisible by products of binomials of the form 1 + q i , the quotients being q-analogs of the reduced tangent numbers having an explicit combinatorial interpretation.

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Section: Since Euler Both Eulerian Polynomialssupporting

confidence: 52%

Doubloons and New Q-Tangent Numbers

Foata¹,

Han²

2009

The Quarterly Journal of Mathematics

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“…as was done by Carlitz [Ca54] in his seminal paper, thereby entering the q-series environment initiated by Heine [He1847]. Also see Gasper and Rahman [GR90].…”

Section: The Carlitz Q-eulerian Polynomialsmentioning

confidence: 97%

“…Our purpose in the sequel is to present Euler's discovery by making a contemporary reading of his calculation. Two centuries after Euler the Eulerian polynomials were given an extension in the algebra of the q-series, thanks to Carlitz [Ca54]. Our intention is also to discuss some aspects of that q-extension with a short detour to contemporary works in Combinatorics.…”

Section: Introductionmentioning

confidence: 99%

Eulerian Polynomials: From Euler’s Time to the Present

Foata¹

2010

The Legacy of Alladi Ramakrishnan in the Mathematical Sciences

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Abstract. The polynomials commonly called "Eulerian" today have been introduced by Euler himself in his famous book "Institutiones calculi differentialis cum eius usu in analysi finitorum ac Doctrina serierum" (chap. VII), back in 1755. They have been since thoroughly studied, extended, applied. The purpose of the present paper is to go back to Euler's memoir, find out his motivation and reproduce his derivation, surprisingly partially forgotten. The rebirth of those polynomials in a q-environment is due to Carlitz two centuries after Euler. A brief overview of Carlitz's method is given, as well as a short presentation of combinatorial works dealing with natural extensions of the classical Eulerian polynomials.

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“…These q-Eulerian numbers differ from those of Carlitz [3]. In fact, Carlitz's q-Eulerian numbers A,s (q) can be defined as…”

Section: Proposition Let (L A) Be An Ss-lattice Of Rank N and Let mentioning

confidence: 98%