<abstract xmlns="http://www.w3.org/1999/xhtml">We introduce new generalizations of the Bernoulli, Euler, and Genocchi polynomials and numbers based on the Carlitz-Tsallis degenerate exponential function. Also, we present generalizations of some familiar identities and connection between these types of Bernoulli, Euler, and Genocchi polynomials. Moreover, we establish new analogues of the Euler identity for degenerate Bernoulli polynomials and numbers.</abstract>

New degenerate Bernoulli, Euler, and Genocchi polynomials

<abstract xmlns="http://www.w3.org/1999/xhtml">Let 𝒯(<italic>k</italic>)<italic>n</italic> denote the set of <italic>k</italic>-Stirling permutations having <italic>n</italic> distinct letters. Here, we consider the number of steps required (i.e., <italic>pushes</italic>) to rearrange the letters of a member of 𝒯(<italic>k</italic>)<italic>n</italic> so that they occur in non-decreasing order. We find recurrences for the joint distribution on 𝒯(<italic>k</italic>)<italic>n</italic> for the statistics recording the number of levels (i.e., occurrences of equal adjacent letters) and pushes. When <italic>k</italic> = 2, an explicit formula for the ordinary generating function of this distribution is also found. In order to do so, we determine the <italic>LU</italic>-decomposition of a certain infinite matrix having polynomial entries which enables one to compute explicitly the inverse matrix.</abstract>

Counting Stirling permutations by number of pushes

<abstract xmlns="http://www.w3.org/1999/xhtml">The area of S-Motzkin paths (bijective to ternary trees) is calculated using the kernel method by enumerating these (partial) paths with fixed end-point resp. starting point.</abstract>

Enumeration of S-Motzkin paths from left to right and from right to left: a kernel method approach

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Pure Mathematics and Applications

Volume 29 (2020): Issue 1 (December 2020)

New degenerate Bernoulli, Euler, and Genocchi polynomials

Counting Stirling permutations by number of pushes

Enumeration of S-Motzkin paths from left to right and from right to left: a kernel method approach

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