The basic theory of difference equations is based on the difference operator Δ defined as \mathrm{\Delta}u(k)=u(k+1)-u(k), k\in \mathbb{N}=\{0,1,2,3,\dots \}, which allows the recursive computation of solutions. Later, the following definition was suggested for {\mathrm{\Delta}}_{\ell} by [1–3] and [4]:

{\mathrm{\Delta}}_{\ell}u(k)=u(k+\ell )-u(k),\phantom{\rule{1em}{0ex}}k\in \mathbb{R},\ell \in \mathbb{R}-\{0\};

(2)

however, no significant progress took place on this line. Recently, equation (2) was reconsidered and its inverse was defined by {\mathrm{\Delta}}_{\ell}^{-1}, and many interesting results in applications such as in number theory as well as in fluid dynamics were obtained; see, for example, [5]. By extending the study for sequences of complex numbers and *ℓ* to be real, some new qualitative properties like rotatory, expanding, shrinking, spiral and weblike were studied for the solutions of difference equations involving {\mathrm{\Delta}}_{\ell}. The {\ell}_{2} and {c}_{0} solutions of the second-order difference equation of (1) when \ell =1 were discussed in [6] and further generalized in [7]. In this paper, we discuss some applications of {\mathrm{\Delta}}_{\ell} in the finite and infinite series of number theory.

In this section, we present some of the preliminary definitions and results which will be useful for future discussion. The following definitions were held in [5] and [8], respectively.

**Definition 1.1** Let u(k), k\in [0,\mathrm{\infty}), be a real- or complex-valued function and \ell \in (0,\mathrm{\infty}). Then the generalized difference operator {\mathrm{\Delta}}_{\ell} is defined as

{\mathrm{\Delta}}_{\ell}u(k)=u(k+\ell )-u(k).

(3)

Then the inverse of {\mathrm{\Delta}}_{\ell} denoted by {\mathrm{\Delta}}_{\ell}^{-1} is defined as follows: If

{\mathrm{\Delta}}_{\ell}v(k)=u(k),\phantom{\rule{1em}{0ex}}\text{then}v(k)={\mathrm{\Delta}}_{\ell}^{-1}u(k)+{c}_{j},

(4)

where {c}_{j} is a constant for all k\in {\mathbb{N}}_{\ell}(j), j=k-[\frac{k}{\ell}]\ell. If {lim}_{k\to \mathrm{\infty}}u(k)=0, then we can take {c}_{j}=0. Further, the generalized polynomial factorial for \ell >0 is defined as

{k}_{\ell}^{(n)}=k(k-\ell )(k-2\ell )\cdots (k-(n-1)\ell ).

(5)

The following lemmas were proved in [9] and [10], respectively.

**Lemma 1.2** (Product formula)

*Let* u(k) *and* v(k), k\in [0,\mathrm{\infty}), *be any two real*-*valued functions*. *Then*

\begin{array}{rcl}{\mathrm{\Delta}}_{\ell}\{u(k)v(k)\}& =& u(k+\ell ){\mathrm{\Delta}}_{\ell}v(k)+v(k){\mathrm{\Delta}}_{\ell}u(k)\\ =& v(k+\ell ){\mathrm{\Delta}}_{\ell}u(k)+u(k){\mathrm{\Delta}}_{\ell}v(k).\end{array}

(6)

**Lemma 1.3** *Let* \ell >0, n\in \mathbb{N}(2), k\in (\ell ,\mathrm{\infty}) *and* {k}_{\ell}^{(n)}\ne 0. *Then*

{\mathrm{\Delta}}_{\ell}^{-1}\frac{1}{{k}_{\ell}^{(n)}}=\frac{-1}{(n-1)\ell {(k-\ell )}_{\ell}^{(n-1)}}+{c}_{j}.

(7)

**Definition 1.4** A function u(k), k\in [a,\mathrm{\infty}), is said to be in the {\ell}_{2(\ell )}-space if

\sum _{\gamma =0}^{\mathrm{\infty}}|u(a+j+\gamma \ell ){|}^{2}<\mathrm{\infty}\phantom{\rule{1em}{0ex}}\text{for all}j\in [0,\ell ).

(8)

If {lim}_{r\to \mathrm{\infty}}|u(a+j+r\ell )|=0 for all j\in [0,\ell ), then u(k) is said to be in the {c}_{0(\ell )}-space.

In what follows, we have the summation formula for finite and infinite series.

**Lemma 1.5** *If a real*-*valued function* u(k) *is defined for all* k\in [0,\mathrm{\infty}), *then*

{\mathrm{\Delta}}_{\ell}^{-1}u(k)=\sum _{r=1}^{[\frac{k}{\ell}]}u(k-r\ell )+{c}_{j},

(9)

*where* {c}_{j} *is a constant for all* k\in {\mathbb{N}}_{\ell}(j), j=k-[\frac{k}{\ell}]\ell. *Since* [0,\mathrm{\infty})={\bigcup}_{0\le j<\ell}{N}_{\ell}(j), *each complex number* {c}_{j} (0\le j<\ell) *is called an initial value of* k\in {N}_{\ell}(j). *Usually*, *each initial value* {c}_{j} *is taken from any one of the values* u(j), u(j+\ell ), u(j+2\ell ), *etc*. *Further*, *if* {lim}_{k\to \mathrm{\infty}}u(k)=0 *and* \ell >0, *then*

{\mathrm{\Delta}}_{\ell}^{-1}u(k)=-\sum _{r=0}^{\mathrm{\infty}}u(k+r\ell ).

(10)

*Proof* Assume z(k)={\sum}_{r=0}^{\mathrm{\infty}}u(k+r\ell ). Then

{\mathrm{\Delta}}_{\ell}z(k)=\sum _{r=0}^{\mathrm{\infty}}u(k+\ell +r\ell )-\sum _{r=0}^{\mathrm{\infty}}u(k+r\ell )=-u(k).

Now, the proof follows from {lim}_{k\to \mathrm{\infty}}u(k)=0 and Definition 1.1. □

The next lemma is an expansion of Lemma 1.5 and its proof is straightforward.

**Lemma 1.6** *If* {lim}_{k\to \mathrm{\infty}}u(k)=0 *and* \ell >0, *then*

{\mathrm{\Delta}}_{\ell}^{-2}u(k)=\sum _{{r}_{1}=0}^{\mathrm{\infty}}\sum _{{r}_{2}=0}^{\mathrm{\infty}}u(k+{r}_{1}\ell +{r}_{2}\ell ).

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**Theorem 1.7** *Let* n\in \mathbb{N}(2), k\in (0,\mathrm{\infty}) *such that* {k}_{\ell}^{(n)}\ne 0. *Then*

\sum _{r=0}^{\mathrm{\infty}}\frac{1}{{(k+r\ell )}_{\ell}^{(n)}}=\frac{1}{(n-1)\ell {(k-\ell )}_{\ell}^{(n-1)}}.

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*Proof* The proof follows from Lemma 1.3 and Lemma 1.5 by taking u(k)=\frac{1}{{k}_{\ell}^{(n)}} and {c}_{j}=0. □

**Corollary 1.8** *Let* k\in (\ell ,\mathrm{\infty}) *and* \ell \in (0,\mathrm{\infty}). *Then*

\sum _{r=0}^{\mathrm{\infty}}\frac{1}{(k+r\ell )(k+r\ell -\ell )}=\frac{1}{\ell (k-\ell )}.

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*Proof* Since {\mathrm{\Delta}}_{\ell}^{-1}\frac{1}{k(k-\ell )}=\frac{-1}{\ell (k-\ell )}, the proof follows from Theorem 1.7 by taking n=2. □