Catalan–Qi Numbers, Series Involving the Catalan–Qi Numbers and a Hankel Determinant Evaluation

Several Series Identities Involving the Catalan Numbers

In the paper, the authors discover several series identities involving the Catalan numbers, the Catalan function, the Riemanian zeta function, and the alternative Hurwitz zeta function.

A series involving Catalan numbers: Proofs and demonstrations

Sequences and Series Involving the Sequence of Composite Numbers

A Catalan-Hankel Determinant Evaluation

Let Ck = ( 2k k ) /(k+1) denote the k-th Catalan number and put ak(x) = Ck+Ck−1x+· · ·+C0x . Define the (n+1)×(n+1) Hankel determinant by setting Hn(x) = det[ai+j(x)]0≤i,j≤n. Even though Hn(x) does not admit a product form evaluation for arbitrary x, the recently introduced technique of γ-operators is applicable. We illustrate this technique by evaluating this Hankel determinant as Hn(x) = n ∑

Generalized Catalan Numbers and Generalized Hankel Transformations

so that an is the sum of the n th and n + 1 Catalan numbers. Then the Hankel transform of {a0, a1, a2, . . .} begins 2, 5, 13, 34, . . .. Layman first conjectured in the On-Line Encyclopedia of Integer Sequences ([4], see sequence A001906) that this sequence consists of every other Fibonacci number, and subsequently Cvetković, Rajković and Ivković [2] proved this conjecture. The current paper a...