On the Reciprocal Sums of Products of Balancing and Lucas-Balancing Numbers

Recently Panda et al. obtained some identities for the reciprocal sums of balancing and Lucas-balancing numbers. In this paper, we derive general identities related to reciprocal sums of products of two balancing numbers, products of two Lucas-balancing numbers and products of balancing and Lucas-balancing numbers. The method of this paper can also be applied to even-indexed and odd-indexed Fibonacci, Lucas, Pell and Pell–Lucas numbers.

On the Reciprocal Sums of Products of Two Generalized Bi-Periodic Fibonacci Numbers

This paper concerns the properties of the generalized bi-periodic Fibonacci numbers {Gn} generated from the recurrence relation: Gn=aGn−1+Gn−2 (n is even) or Gn=bGn−1+Gn−2 (n is odd). We derive general identities for the reciprocal sums of products of two generalized bi-periodic Fibonacci numbers. More precisely, we obtain formulas for the integer parts of the numbers ∑k=n∞(a/b)ξ(k+1)GkGk+m−1,m=0,2,4,⋯, and ∑k=n∞1GkGk+m−1,m=1,3,5,⋯.

On the Chebyshev Polynomials and Some of Their Reciprocal Sums

In this paper, we utilize the mathematical induction, the properties of symmetric polynomial sequences and Chebyshev polynomials to study the calculating problems of a certain reciprocal sums of Chebyshev polynomials, and give two interesting identities for them. These formulae not only reveal the close relationship between the trigonometric function and the Riemann ζ-function, but also generalized some existing results. At the same time, an error in an existing reference is corrected.

On the Sum of Reciprocal of Polynomial Applied to Higher Order Recurrences

Recently a lot of papers have been devoted to partial infinite reciprocal sums of a higher-order linear recursive sequence. In this paper, we continue this program by finding a sequence which is asymptotically equivalent to partial infinite sums, including a reciprocal of polynomial applied to linear higher order recurrences.