From q-Stirling numbers to the ordered multiset partitions: A viewpoint from vincular patterns

Chen, Joanna N.; Fu, Shishuo

doi:10.1016/j.aam.2019.06.007

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Certain Identities Involving the (P Q)-binomial Coefficients1

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2020

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“…We use ( 51) in ( 56) to express (p, q)-analogue of Bernstein polynomials (55) in the following double series like the form of the q-analogue of Bernstein polynomials which is obtained by substituting ( 27) for ν h 0 in (25) (see also ([72], Propostion 1)):…”

Section: Certain Identities Involving the (P Q)-binomial Coefficientsmentioning

confidence: 99%

Certain Identities Associated with (p,q)-Binomial Coefficients and (p,q)-Stirling Polynomials of the Second Kind

2020

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The q-Stirling numbers (polynomials) of the second kind have been investigated and applied in a variety of research subjects including, even, the q-analogue of Bernstein polynomials. The (p,q)-Stirling numbers (polynomials) of the second kind have been studied, particularly, in relation to combinatorics. In this paper, we aim to introduce new (p,q)-Stirling polynomials of the second kind which are shown to be fit for the (p,q)-analogue of Bernstein polynomials. We also present some interesting identities involving the (p,q)-binomial coefficients. We further discuss certain vanishing identities associated with the q-and (p,q)-Stirling polynomials of the second kind.

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Section: Certain Identities Involving the (P Q)-binomial Coefficientsmentioning

confidence: 99%