Melham discovered the Fibonacci identity $F_{n+1}{F}_{n+2}{F}_{n+6}-F{³}_{n+3}=(-1)ⁿFₙ$. He then considered the generalized sequence Wₙ where W₀ = a, W₁ = b, and $Wₙ=p{W}_{n-1}+q{W}_{n-2}$ and a, b, p and q are integers and q ≠ 0. Letting e = pab - qa² - b², he proved the following identity: $W_{n+1}{W}_{n+2}{W}_{n+6}-W{³}_{n+3}=e{q}^{n+1}(p³W_{n+2}-q²W_{n+1})$. There are similar differences of products of Fibonacci numbers, like this one discovered by Fairgrieve and Gould: $FₙF_{n+4}{F}_{n+5}-F{³}_{n+3}={(-1)}^{n+1}{F}_{n+6}$. We prove similar identities. For example, a generalization of Fairgrieve and Gould’s identity is $WₙW_{n+4}{W}_{n+5}-W{³}_{n+3}=eqⁿ(p³W_{n+4}-q{W}_{n+5})$.

In this paper, by considering higher-order degenerate Bernoulli and Euler polynomials which were introduced by Carlitz, we investigate some properties of mixed-type of those polynomials. In particular, we give some identities of mixed-type degenerate special polynomials which are derived from the fermionic integrals on Zp and the bosonic integrals on Zp.

We display several infinite products with interesting continued fraction expansions. Specifically, for various small values of $k$ necessarily excluding $k=3$ since that case cannot occur, we display infinite products in the field of formal power series whose truncations yield their every $k$-th convergent.

We find the sum of series of the form $$\sum _{i=1}^{\infty }\frac{f\left(i\right)}{i^r}$$ for some special functions $f$. The above series is a generalization of the Riemann zeta function. In particular, we take $f$ as some values of Hurwitz zeta functions, harmonic numbers, and combination of both. These generalize some of the results given in Mező’s paper (2013). We use multiple zeta theory to prove all results. The series sums we have obtained are in terms of Bernoulli numbers and powers of $\pi$.

In this paper we consider two parameters generalization of the Fibonacci numbers and Pell numbers, named as the $(k,p)$-Fibonacci numbers. We give some new interpretations of these numbers. Moreover using these interpretations we prove some identities for the $(k,p)$-Fibonacci numbers.