Minors of a class of Riordan arrays related to weighted partial Motzkin paths

On identities involving generalized harmonic, hyperharmonic and special numbers with Riordan arrays

Special Matrices ◽

10.1515/spma-2020-0111 ◽

2021 ◽

Vol 9 (1) ◽

pp. 22-30

Author(s):

Sibel Koparal ◽

Neşe Ömür ◽

Ömer Duran

Keyword(s):

Stirling Numbers ◽

Bernoulli Numbers ◽

Riordan Arrays ◽

Riordan Array

Abstract In this paper, by means of the summation property to the Riordan array, we derive some identities involving generalized harmonic, hyperharmonic and special numbers. For example, for n ≥ 0, ∑ k = 0 n B k k ! H ( n . k , α ) = α H ( n + 1 , 1 , α ) - H ( n , 1 , α ) , \sum\limits_{k = 0}^n {{{{B_k}} \over {k!}}H\left( {n.k,\alpha } \right) = \alpha H\left( {n + 1,1,\alpha } \right) - H\left( {n,1,\alpha } \right)} , and for n > r ≥ 0, ∑ k = r n - 1 ( - 1 ) k s ( k , r ) r ! α k k ! H n - k ( α ) = ( - 1 ) r H ( n , r , α ) , \sum\limits_{k = r}^{n - 1} {{{\left( { - 1} \right)}^k}{{s\left( {k,r} \right)r!} \over {{\alpha ^k}k!}}{H_{n - k}}\left( \alpha \right) = {{\left( { - 1} \right)}^r}H\left( {n,r,\alpha } \right)} , where Bernoulli numbers Bn and Stirling numbers of the first kind s (n, r).

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Standard Young tableaux in a (2,1)-hook and Motzkin paths

Discrete Mathematics ◽

10.1016/j.disc.2021.112395 ◽

2021 ◽

Vol 344 (7) ◽

pp. 112395

Author(s):

Rosena R.X. Du ◽

Jingni Yu

Keyword(s):

Young Tableaux ◽

Standard Young Tableaux ◽

Motzkin Paths

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On the Square Root of a Bell Matrix

Results in Mathematics ◽

10.1007/s00025-021-01356-y ◽

2021 ◽

Vol 76 (1) ◽

Author(s):

Donatella Merlini

Keyword(s):

Power Series ◽

Closed Form ◽

Formal Power Series ◽

Computational Method ◽

Square Root ◽

Riordan Arrays ◽

Formal Power

AbstractIn the context of Riordan arrays, the problem of determining the square root of a Bell matrix $$R={\mathcal {R}}(f(t)/t,\ f(t))$$ R = R ( f ( t ) / t , f ( t ) ) defined by a formal power series $$f(t)=\sum _{k \ge 0}f_kt^k$$ f ( t ) = ∑ k ≥ 0 f k t k with $$f(0)=f_0=0$$ f ( 0 ) = f 0 = 0 is presented. It is proved that if $$f^\prime (0)=1$$ f ′ ( 0 ) = 1 and $$f^{\prime \prime }(0)\ne 0$$ f ″ ( 0 ) ≠ 0 then there exists another Bell matrix $$H={\mathcal {R}}(h(t)/t,\ h(t))$$ H = R ( h ( t ) / t , h ( t ) ) such that $$H*H=R;$$ H ∗ H = R ; in particular, function h(t) is univocally determined by a symbolic computational method which in many situations allows to find the function in closed form. Moreover, it is shown that function h(t) is related to the solution of Schröder’s equation. We also compute a Riordan involution related to this kind of matrices.

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Some Properties of the(p,q)-Fibonacci and(p,q)-Lucas Polynomials

Journal of Applied Mathematics ◽

10.1155/2012/264842 ◽

2012 ◽

Vol 2012 ◽

pp. 1-18 ◽

Cited By ~ 6

Author(s):

GwangYeon Lee ◽

Mustafa Asci

Keyword(s):

Generating Functions ◽

Combinatorial Identities ◽

Fibonacci Polynomials ◽

Lucas Polynomials ◽

Pascal Matrix ◽

Riordan Arrays ◽

Combinatorial Sums

Riordan arrays are useful for solving the combinatorial sums by the help of generating functions. Many theorems can be easily proved by Riordan arrays. In this paper we consider the Pascal matrix and define a new generalization of Fibonacci polynomials called(p,q)-Fibonacci polynomials. We obtain combinatorial identities and by using Riordan method we get factorizations of Pascal matrix involving(p,q)-Fibonacci polynomials.

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