Simplicial d-Polytopic Numbers Defined on Lucas Sequences

Ronald Orozco López Departamento de Matemáticas Universidad de los Andes Bogotá, 111711 Colombia

                   rj.orozco@uniandes.edu.co

Abstract

This paper introduces the simplicial d-polytopic numbers defined on Lucas sequences. We establish basic identities and find q-identities. Furthermore, we find generating functions for the simplicial d-Lucas-polytopic numbers and for the squares of the Lucas-triangular numbers. Finally, we compute sums of reciprocals of Lucas sequences and Lucas-triangular numbers. We introduce an analogue of the zeta function defined on Lucas sequences.

1 Introduction

There is growing research on analogue sequences of numbers defined in terms of Lucas sequences, including Catalan numbers [1, 5], Bernoulli and Euler polynomials [7], Eulerian numbers [10], among others. In this paper, we define simplicial d-polytopic numbers on Lucas sequences. The advantage of doing so is that we obtain, for free, analogs of Fibonacci, Pell, Jacobsthal, and Mersenne sequences, among others.

The simplicial polytopic numbers [3] are a family of sequences of figurate numbers corresponding to the d-dimensional simplex for each dimension d, where d is a non-negative integer. For d ranging from 1 to 5, we have the following simplicial polytopic numbers, respectively: non-negative numbers ℕ, triangular numbers T_{n}, tetrahedral numbers Te_{n}, pentachoron numbers P_{n} and hexateron numbers H_{n}. A list of the above sets of numbers is as follows:

ℕ = (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, …),
T = (0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, …),
Te = (0, 1, 4, 10, 20, 35, 56, 84, 120, 165, …),
P = (0, 1, 5, 15, 35, 70, 126, 210, 330, 495, 715, …),
H = (0, 1, 6, 21, 56, 126, 252, 462, 792, 1287, …).

The n^{th} simplicial d-polytopic numbers P_{n}^{d} are given by the formula

P_{n}^{d} = \binom{n + d − 1}{d} = \frac{n^{(d)}}{d!} ,

where x^{(d)} = x(x + 1)(x + 2) ⋯ (x + d − 1) is the rising factorial. The generating function of the simplicial d-polytopic numbers is

∑_{n=1}^{∞} \binom{n + d − 1}{d}x^{n} = \frac{x}{(1 − x)^{d+1}} .

In this paper, the n-th simplicial d-Lucas-polytopic number is defined by

\left\{{n + d − 1 \atop d}\right\}_{s,t} = \frac{\{n\}_{s,t}\{n + 1\}_{s,t} ⋯ \{n + d − 1\}_{s,t}}{\{d\}_{s,t}!} ,

where {n}_{s,t} is the Lucas analogue of the positive integer n. We determine basic identities for simplicial Lucas polytopic numbers, especially for Lucas-triangular and Lucas-tetrahedral numbers. These sequences are part of the On-Line Encyclopedia of Integer Sequences [13]. Some known q-identities are found, [12, 14]. We establish generating functions for the simplicial d-Lucas-polytopic numbers \left\{{n+d−1 \atop d}\right\}_{s,t} and for the sequence \left\{{n+1 \atop 2}\right\}_{s,t}^{2}. Finally, we introduce the Lucas-zeta function ζ_{s,t}(z) and find some values for ζ_{s,t}(1). In addition, we calculate reciprocal sums of Lucas sequences and Lucas-triangular numbers.

2 Preliminaries

The Lucas sequences [11] on the parameters s, t are defined by

{n + 2}_{s,t} = s{n + 1}_{s,t} + t{n}_{s,t}

with initial values {0}_{s,t} = 0 and {1}_{s,t} = 1, where s ≠ 0 and t ≠ 0. Below are some important specializations of Lucas sequences.

1. If s = 2, t = −1, then {n}_{2,−1} = n are the positive integers.

2. If s = 1, t = 1, then {n}_{1,1} = F_{n} are the Fibonacci numbers F_{n} = (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …).

3. If s = 2, t = 1, then {n}_{2,1} = P_{n}, where P_{n} are the Pell numbers P_{n} = (0, 1, 2, 5, 12, 29, 70, 169, 408 …).

4. If s = 1, t = 2, then {n}_{1,2} = J_{n}, where J_{n} are the Jacobsthal numbers J_{n} = (0, 1, 1, 3, 5, 11, 21, 43, 85, 171, …).

5. If s = 3, t = −2, then {n}_{3,−2} = M_{n}, where M_{n} = 2^{n} − 1 are the Mersenne numbers M_{n} = (0, 1, 3, 7, 15, 31, 63, 127, 255, …).

6. If s = p + q, t = −pq, then {n}_{p+q,−pq} = [n]_{p,q}, where [n]_{p,q} are the (p, q)-numbers

[n]_{p,q} = (0, 1, [2]_{p,q}, [3]_{p,q}, [4]_{p,q}, [5]_{p,q}, [6]_{p,q}, [7]_{p,q}, [8]_{p,q} …).

If p = 1, we obtain the q-numbers [n]_{q} = 1 + q + q^{2} + q^{3} + ⋯ + q^{n−1}.

7. If s = 2t, t = −1, then {n}_{2t,−1} = U_{n−1}(t), where U_{n}(t) are the Chebyshev polynomials

of the second kind, with U_{−1}(t) = 0. The Lucas constant is the ratio toward which adjacent terms in a Lucas sequence tend. This is the only positive zero of x^{2} − sx − t = 0. We let φ_{s,t} denote this constant, where

φ_{s,t} = \frac{s + \sqrt{s^{2} + 4t}}{2}

and

φ′_{s,t} = s − φ_{s,t} = −\frac{t}{φ_{s,t}} = \frac{s − \sqrt{s^{2} + 4t}}{2}

denote the reciprocal of φ_{s,t}. Some specializations of the constants φ_{s,t} and φ′_{s,t} are 1. If s = 2 and t = −1, then φ_{2,−1} = 1 and φ′_{2,−1} = 1.

2. If s = 1 and t = 1, then φ_{1,1} = φ = \frac{1+\sqrt{5}}{2} and φ′_{1,1} = φ′ = \frac{1−\sqrt{5}}{2}.

3. If s = 2 and t = 1, then φ_{2,1} = 1 + \sqrt{2} and φ′_{2,1} = 1 − \sqrt{2}.

4. If s = 1 and t = 2, then φ_{1,2} = 2 and φ′_{1,2} = −1.

5. If s = 3 and t = −2, then φ_{3,−2} = 2 and φ′_{3,−2} = 1.

6. If s = p + q and t = −pq, then φ_{p+q,−pq} = p and φ′_{p+q,−pq} = q.

7. If s = 2t and t = −1, then φ_{2t,−1} = \frac{t+\sqrt{t^{2}−1}}{2} and φ′_{2t,−1} = \frac{t−\sqrt{t^{2}−1}}{2}.

The Lucasnomial coefficients are defined by

\left\{{n \atop k}\right\}_{s,t} = \frac{\{n\}_{s,t}!}{\{k\}_{s,t}!\{n − k\}_{s,t}!},

where {n}_{s,t}! = {1}_{s,t}{2}_{s,t} ⋯ {n}_{s,t}. The Lucasnomial coefficients satisfy the following Pascal recurrence relationships. For 1 ≤ k ≤ n − 1 we have

\left\{{n + 1 \atop k}\right\}_{s,t} = φ_{s,t}^{k}\left\{{n \atop k}\right\}_{s,t} + φ_{s,t}^{′(n+1−k)}\left\{{n \atop k − 1}\right\}_{s,t},  (1)
= φ_{s,t}^{′(k)}\left\{{n \atop k}\right\}_{s,t} + φ_{s,t}^{n+1−k}\left\{{n \atop k − 1}\right\}_{s,t}.  (2)

A proof of the above identities was provided by Corcino [2].

Set s, t ∈ ℝ, with s ≠ 0 and t ≠ 0. If s^{2} + 4t ≠ 0, we define the Lucas-derivative D_{s,t} of the function f(x) as

(D_{s,t}f)(x) = \begin{cases}\frac{f(φ_{s,t}x)−f(φ′_{s,t}x)}{(φ_{s,t}−φ′_{s,t})x}, & if x ≠ 0; \\ f′(0), & if x = 0, \end{cases}

provided f(x) is differentiable at x = 0. If s^{2} + 4t = 0, with t < 0, we define the Lucasderivative of the function f(x) as

(D_{±2i\sqrt{t},t}f)(x) = f′(±i\sqrt{t}x).

The Lucas-derivative D_{s,t} fulfills the following properties.

• Linearity:

D_{s,t}(αf + βg) = αD_{s,t}f + βD_{s,t}g.

• Product rules:

D_{s,t}(f(x)g(x)) = f(φ_{s,t}x)D_{s,t}g(x) + g(φ′_{s,t}x)D_{s,t}f(x),

and

D_{s,t}(f(x)g(x)) = f(φ′_{s,t}x)D_{s,t}g(x) + g(φ_{s,t}x)D_{s,t}f(x).

• Quotient rules:

D_{s,t}(\frac{f(x)}{g(x)}) = \frac{g(φ_{s,t}x)D_{s,t}f(x) − f(φ_{s,t}x)D_{s,t}g(x)}{g(φ_{s,t}x)g(φ′_{s,t}x)},

and

D_{s,t}(\frac{f(x)}{g(x)}) = \frac{g(φ′_{s,t}x)D_{s,t}f(x) − f(φ′_{s,t}x)D_{s,t}g(x)}{g(φ_{s,t}x)g(φ′_{s,t}x)}.

Define the n-th Lucas-derivative of the function f(x) recursively as

D_{s,t}^{n}f(x) = D_{s,t}(D_{s,t}^{n}f(x)).

3 d-Lucas-polytopic numbers

3.1 Definition and basic properties

Definition 1. The n-th simplicial d-Lucas-polytopic number is defined by

\left\{{n + d − 1 \atop d}\right\}_{s,t} = \frac{\{n\}_{s,t}\{n + 1\}_{s,t} ⋯ \{n + d − 1\}_{s,t}}{\{d\}_{s,t}!} .

The Lucas analogues of the triangular numbers, tetrahedral numbers, pentachoron numbers, and hexateron numbers are

T_{s,t} = \{T_{n}(s, t) = \left\{{n + 1 \atop 2}\right\}_{s,t} : n ≥ 0\},

Te_{s,t} = \{Te_{n}(s, t) = \left\{{n + 2 \atop 3}\right\}_{s,t} : n ≥ 0\},

P_{s,t} = \{P_{n}(s, t) = \left\{{n + 3 \atop 4}\right\}_{s,t} : n ≥ 0\},

H_{s,t} = \{H_{n}(s, t) = \left\{{n + 4 \atop 5}\right\}_{s,t} : n ≥ 0\}.

From the Pascal recurrence in Eqs. (1) and (2) we have

\left\{{n + d \atop d}\right\}_{s,t} = φ_{s,t}^{d}\left\{{n + d − 1 \atop d}\right\}_{s,t} + φ_{s,t}^{′n}\left\{{n + d − 1 \atop d − 1}\right\}_{s,t},  (3)
= φ_{s,t}^{′d}\left\{{n + d − 1 \atop d}\right\}_{s,t} + φ_{s,t}^{n}\left\{{n − d − 1 \atop d − 1}\right\}_{s,t}.  (4)

It is a well-known fact that the sum of the first n terms of a sequence of d-polytopic numbers is the n-th term of a sequence of (d + 1)-polytopic numbers, i.e,

∑_{k=1}^{n} P_{n}^{d} = P_{n}^{d+1}.

We then obtain the Lucas analogue of the above formula.

Theorem 2. For all n ≥ 1, the sum of simplicial d-Lucas-polytopic numbers is

\left\{{n + d \atop d + 1}\right\}_{s,t} = ∑_{k=1}^{n} φ_{s,t}^{(d+1)(n−k)}φ_{s,t}^{′(k−1)}\left\{{k + d − 1 \atop d}\right\}_{s,t},
\left\{{n + d \atop d + 1}\right\}_{s,t} = ∑_{k=1}^{n} φ_{s,t}^{′(d+1)(n−k)}φ_{s,t}^{k−1}\left\{{k + d − 1 \atop d}\right\}_{s,t}.

Proof. The proof is by induction on n. When n = 1, then \left\{{d+1 \atop d+1}\right\}_{s,t} = \left\{{d \atop d}\right\}_{s,t}. For n = 2, it follows that

\left\{{d + 2 \atop d + 1}\right\}_{s,t} = φ_{s,t}^{d+1}\left\{{d \atop d}\right\}_{s,t} + φ′_{s,t}\left\{{d + 1 \atop d}\right\}_{s,t}.

Suppose the statement is true for n and we prove the statement for n + 1. We have

\left\{{n + d + 1 \atop d + 1}\right\}_{s,t} = φ_{s,t}^{d+1}\left\{{n + d \atop d + 1}\right\}_{s,t} + φ_{s,t}^{′n}\left\{{n + d \atop d}\right\}_{s,t}

= φ_{s,t}^{d+1} ∑_{k=1}^{n} φ_{s,t}^{(d+1)(n−k)}φ_{s,t}^{′(k−1)}\left\{{k + d − 1 \atop d}\right\}_{s,t} + φ_{s,t}^{′n}\left\{{n + d \atop d}\right\}_{s,t}

= ∑_{k=1}^{n} φ_{s,t}^{(d+1)(n+1−k)}φ_{s,t}^{′(k−1)}\left\{{k + d − 1 \atop d}\right\}_{s,t} + φ_{s,t}^{′n}\left\{{n + d \atop d}\right\}_{s,t}

= ∑_{k=1}^{n+1} φ_{s,t}^{(d+1)(n+1−k)}φ_{s,t}^{′(k−1)}\left\{{k + d − 1 \atop d}\right\}_{s,t}.

The proof is reached.

From Theorem 2 we obtain the following known result about q-binomial coefficients:

\left[{n + d \atop d + 1}\right]_{q} = ∑_{k=1}^{n} q^{k−1}\left[{k + d − 1 \atop d}\right]_{q} = ∑_{k=1}^{n} q^{(d+1)(n−k)}\left[{k + d − 1 \atop d}\right]_{q}.

4 Some specializations

4.1 Lucas-triangular numbers

Some specializations of Lucas-triangular numbers are

\left\{{n + 1 \atop 2}\right\}_{1,1} = F_{n}F_{n+1} = (0, 1, 2, 6, 15, 40, 104, 273, …),

\left\{{n + 1 \atop 2}\right\}_{2,1} = \frac{1}{2}P_{n}P_{n+1} = (0, 1, 5, 30, 174, 1015, 5915, …),
\left\{{n + 1 \atop 2}\right\}_{1,2} = J_{n}J_{n+1} = (0, 1, 2, 6, 15, 55, 231, 903, 3655, …),
\left\{{n + 1 \atop 2}\right\}_{3,−2} = \frac{1}{3}(2^{n} − 1)(2^{n+1} − 1)
= (0, 1, 7, 35, 155, 651, 2667, 10795, 43435, 174251, …).

The \left\{{n+1 \atop 2}\right\}_{1,1}-numbers are known as golden rectangle numbers, (\underline{A001654} in [13]). The \left\{{n+1 \atop 2}\right\}_{2,1}-numbers may be called Pell triangles, (\underline{A084158} in [13]). The \left\{{n+1 \atop 2}\right\}_{1,2}-numbers are known as Jacobsthal oblong numbers, (\underline{A084175} in [13]). The \left\{{n+1 \atop 2}\right\}_{3,−2}-numbers are the Gaussian binomial coefficients \left[{n \atop 2}\right]_{q} for q = 2, (\underline{A006095} in [13]).

From Eqs. (3) and (4), and Theorem 2, we obtain

\left\{{n + 2 \atop 2}\right\}_{s,t} = φ_{s,t}^{2}\left\{{n + 1 \atop 2}\right\} + φ_{s,t}^{′n}\{n + 1\}_{s,t},
\left\{{n + 2 \atop 2}\right\}_{s,t} = φ_{s,t}^{′2}\left\{{n + 1 \atop 2}\right\}_{s,t} + φ_{s,t}^{n}\{n + 1\}_{s,t},
\left\{{n + 1 \atop 2}\right\}_{s,t} = ∑_{k=1}^{n} φ_{s,t}^{2(n−k)}φ_{s,t}^{′(k−1)}\{k\}_{s,t},
\left\{{n + 1 \atop 2}\right\}_{s,t} = ∑_{k=1}^{n} φ_{s,t}^{′2(n−k)}φ_{s,t}^{k−1}\{k\}_{s,t}.

From the above identities, we obtain the identity of Warnaar [14]

\left[{n + 1 \atop 2}\right]_{q} = ∑_{k=1}^{n} \frac{1 − q^{k}}{1 − q} q^{2(n−k)} = ∑_{k=1}^{n} q^{k−1} \frac{1 − q^{k}}{1 − q} .

Theorem 3. For all n ∈ ℕ, 1. The Lucas analogue of the identity T_{n} + T_{n−1} = n^{2} is

\left\{{n + 1 \atop 2}\right\}_{s,t} = t\left\{{n \atop 2}\right\}_{s,t} + \{n\}^{2}.

2. Another Lucas analogue of the identity T_{n} + T_{n−1} = n^{2} is

φ_{s,t}^{′(n−1)}\left\{{n + 1 \atop 2}\right\}_{s,t} + φ_{s,t}^{n+1}\left\{{n \atop 2}\right\}_{s,t} = \frac{⟨n⟩_{s,t}}{\{2\}_{s,t}} \{n\}_{s,t}^{2},

where ⟨n⟩_{s,t} = φ_{s,t}^{n} + φ_{s,t}^{′n}.

3. The Lucas analogue of the alternating sum squares T_{n} = ∑_{k=1}^{n}(−1)^{n−k}k^{2} is

\left\{{n + 1 \atop 2}\right\}_{s,t} = ∑_{k=1}^{n} t^{n−k}\{k\}_{s,t}^{2}.

Proof. We have

\left\{{n + 1 \atop 2}\right\}_{s,t} = \frac{\{n\}_{s,t}\{n + 1\}_{s,t}}{\{2\}_{s,t}}

= \frac{\{n\}_{s,t}(s\{n\}_{s,t} + t\{n − 1\}_{s,t})}{\{2\}_{s,t}}

= s\frac{\{n\}_{s,t}^{2}}{s} + t\frac{\{n\}_{s,t}\{n − 1\}_{s,t}}{\{2\}_{s,t}}

= \{n\}_{s,t}^{2} + t\left\{{n \atop 2}\right\}_{s,t}.

Statement 2 is proved as follows:

φ_{s,t}^{n+1}\left\{{n \atop 2}\right\}_{s,t} + φ_{s,t}^{′(n−1)}\left\{{n + 1 \atop 2}\right\}_{s,t} = φ_{s,t}^{n+1} \frac{\{n\}_{s,t}\{n − 1\}_{s,t}}{\{2\}_{s,t}} + φ_{s,t}^{′(n−1)} \frac{\{n\}_{s,t}\{n + 1\}_{s,t}}{\{2\}_{s,t}}

= \frac{\{n\}_{s,t}}{\{2\}_{s,t}} (φ_{s,t}^{n+1}\{n − 1\}_{s,t} + φ_{s,t}^{′(n−1)})\{n + 1\}_{s,t})

= \frac{\{n\}_{s,t}}{\{2\}_{s,t}} \{2n\}_{s,t}

= \frac{⟨n⟩_{s,t}}{\{2\}_{s,t}} \{n\}_{s,t}^{2}.

By iterating statement (1), we obtain statement (3). If we set s = 1 + q, and t = −q in the previous theorem, then

\left[{n + 2 \atop 2}\right]_{q} = −q\left[{n + 1 \atop 2}\right]_{q} + (\frac{1 − q^{n+1}}{1 − q} )^{2}  (5)

and

\left[{n \atop 2}\right]_{q} + q^{n−1}\left[{n + 1 \atop 2}\right]_{q} = \frac{1 + q^{n}}{1 + q} [n]_{q}^{2}.

By iterating Eq. (5), we obtain a result due to Schlosser [12]:

\left[{n + 1 \atop 2}\right]_{q} = ∑_{k=1}^{n}(−q)^{n−k}(\frac{1 − q^{k}}{1 − q} )^{2}.

Theorem 4. For all n ∈ ℕ,

\left\{{n + 2 \atop 2}\right\}_{s,t}^{2} − t^{2}\left\{{n + 1 \atop 2}\right\}_{s,t}^{2} = (\frac{\{n + 2\}_{s,t} + t\{n\}_{s,t}}{s} )\{n + 1\}_{s,t}^{3}.  (6)

Proof. From Theorem 3, we have that

\left\{{n + 2 \atop 2}\right\}_{s,t}^{2} = (t\left\{{n + 1 \atop 2}\right\}_{s,t} + \{n + 1\}_{s,t}^{2})^{2}

= t^{2}\left\{{n + 1 \atop 2}\right\}_{s,t}^{2} + 2t\left\{{n + 1 \atop 2}\right\}_{s,t}^{2}\{n + 1\}_{s,t}^{2} + \{n + 1\}_{s,t}^{4}

= t^{2}\left\{{n + 1 \atop 2}\right\}_{s,t}^{2} + 2t\frac{\{n\}_{s,t}\{n + 1\}_{s,t}^{3}}{\{2\}_{s,t}} + \{n + 1\}_{s,t}^{4}

= t^{2}\left\{{n + 1 \atop 2}\right\}_{s,t}^{2} + (\frac{2t}{s}\{n\}_{s,t} + \{n + 1\}_{s,t})\{n + 1\}_{s,t}^{3}

= t^{2}\left\{{n + 1 \atop 2}\right\}_{s,t}^{2} + (\frac{t\{n\}_{s,t} + \{n + 2\}_{s,t}}{s} )\{n + 1\}_{s,t}^{3}.

The remainder of the proof follows straightforwardly.

Eq. (6) is the Lucas analogue of the identity

T_{n+1}^{2} − T_{n}^{2} = n^{3}.

From Warnaar [14] we have

\left[{n + 2 \atop 2}\right]_{q}^{2} − q^{2}\left[{n + 1 \atop 2}\right]_{q}^{2} = (\frac{1 − q^{2(n+1)}}{1 − q^{2}} )(\frac{1 − q^{n+1}}{1 − q} )^{2}.

Theorem 5. For all n ∈ ℕ,

∑_{k=1}^{n} t^{2(n−k)}(\frac{\{k + 1\}_{s,t} + t\{k − 1\}_{s,t}}{s} )\{k\}_{s,t}^{3} = \left\{{n + 1 \atop 2}\right\}_{s,t}^{2}.  (7)

Proof.

\left\{{n + 1 \atop 2}\right\}_{s,t}^{2} = (\left\{{n + 1 \atop 2}\right\}_{s,t}^{2} − t^{2}\left\{{n \atop 2}\right\}_{s,t}^{2}) + (t^{2}\left\{{n \atop 2}\right\}_{s,t}^{2} − t^{4}\left\{{n − 1 \atop 2}\right\}_{s,t}^{2})

+ ⋯ + (t^{2n−2}\left\{{2 \atop 2}\right\}_{s,t}^{2} − t^{2n}\left\{{1 \atop 2}\right\}_{s,t}^{2})

= (\frac{\{n + 1\}_{s,t} + t\{n − 1\}_{s,t}}{s} )\{n\}_{s,t}^{3} + t^{2}(\frac{\{n\}_{s,t} + t\{n − 2\}_{s,t}}{s} )\{n − 1\}_{s,t}^{3}+

⋯ + t^{2n−2}(\frac{\{2\}_{s,t}}{s} )\{1\}_{s,t}^{3}

= ∑_{k=1}^{n} t^{2(n−k)}(\frac{\{k + 1\}_{s,t} + t\{k − 1\}_{s,t}}{s} )\{k\}_{s,t}^{3}.

Eq. (7) can be written as

∑_{k=1}^{n} t^{2(n−k)}(\frac{φ_{s,t}^{2k} − φ_{s,t}^{′2k}}{φ_{s,t}^{2} − φ_{s,t}^{′2}} )(\frac{φ_{s,t}^{k} − φ_{s,t}^{′k}}{φ_{s,t} − φ′_{s,t}} )^{2} = ∑_{k=1}^{n} t^{2(n−k)}[k]_{φ^{2},φ^{′2}}[k]_{φ,φ′}^{2} = \left\{{n + 1 \atop 2}\right\}_{s,t}^{2}. (8)

Eq. (7) is the Lucas analogue of the identity

∑_{k=1}^{n} k^{3} = (∑_{k=1}^{n} k)^{2}.

Corollary 6. For all n ∈ ℕ, we have 1. The Fibonacci analogue of the sum of cubes:

∑_{k=1}^{n}(F_{k+1} + F_{k−1})F_{k}^{3} = F_{n}^{2}F_{n+1}^{2}.

2. The Pell analogue of the sum of cubes:

∑_{k=1}^{n}(P_{k+1} + P_{k−1})P_{k}^{3} = \frac{1}{2}P_{n}^{2}P_{n+1}^{2}.

3. The Jacobsthal analogue of the sum of cubes:

∑_{k=1}^{n} 4^{n−k}(J_{k+1} + 2J_{k−1})J_{k}^{3} = J_{n}^{2}J_{n+1}^{2}.

4. The Mersenne analogue of the sum of cubes:

∑_{k=1}^{n} 4^{n−k}(2^{k} + 1)(2^{k} − 1)^{3} = \frac{1}{3}(2^{n} − 1)^{2}(2^{n+1} − 1)^{2}.

5. The q-identity of Warnaar [14]:

∑_{k=1}^{n} q^{2(n−k)}(\frac{1 − q^{2k}}{1 − q^{2}} )(\frac{1 − q^{k}}{1 − q} )^{2} = \left[{n + 1 \atop 2}\right]_{q}^{2}.

4.2 Lucas-tetrahedral numbers

Some specializations of Lucas-tetrahedral numbers are

\left\{{n + 2 \atop 3}\right\}_{1,1} = \frac{1}{2}F_{n}F_{n+1}F_{n+2}
= (0, 1, 3, 15, 60, 260, 1092, 4641, 19635, …),
\left\{{n + 2 \atop 3}\right\}_{2,1} = \frac{1}{10}P_{n}P_{n+1}P_{n+2}
= (1, 12, 174, 2436, 34307, 482664, …),
\left\{{n + 2 \atop 3}\right\}_{1,2} = \frac{1}{3}J_{n}J_{n+1}J_{n+2}
= (0, 1, 5, 55, 385, 3311, 25585, 208335, …),
\left\{{n + 2 \atop 3}\right\}_{3,−2} = \frac{1}{21}(2^{n} − 1)(2^{n+1} − 1)(2^{n+2} − 1)
= (0, 1, 15, 155, 1395, 11811, 97155, …).

The sequences \left\{{n+2 \atop 3}\right\}_{1,1}, \left\{{n+2 \atop 3}\right\}_{2,1}, and \left\{{n+2 \atop 3}\right\}_{3,−2} correspond to \underline{A001655}, \underline{A099930}, and \underline{A006096}, respectively, in [13]. The sequence \left\{{n+2 \atop 3}\right\}_{1,2} has not been previously investigated. From Eqs. (3) and (4), and Theorem 2,

\left\{{n + 3 \atop 3}\right\}_{s,t} = φ_{s,t}^{3}\left\{{n + 2 \atop 3}\right\}_{s,t} + φ_{s,t}^{′n}\left\{{n + 2 \atop 2}\right\}_{s,t},
\left\{{n + 3 \atop 3}\right\}_{s,t} = φ_{s,t}^{′3}\left\{{n + 2 \atop 3}\right\}_{s,t} + φ_{s,t}^{n}\left\{{n + 2 \atop 2}\right\}_{s,t},
\left\{{n + 2 \atop 3}\right\}_{s,t} = ∑_{k=1}^{n} φ_{s,t}^{3(n−k)}φ_{s,t}^{′(k−1)}\left\{{k + 1 \atop 2}\right\}_{s,t},
\left\{{n + 3 \atop 3}\right\}_{s,t} = ∑_{k=1}^{n} φ_{s,t}^{′3(n−k)}φ_{s,t}^{k−1}\left\{{k + 1 \atop 2}\right\}_{s,t}.

Theorem 7. For all n ≥ 0,

\left\{{n + 3 \atop 3}\right\}_{s,t} = st\left\{{n + 2 \atop 3}\right\}_{s,t} + \{n + 1\}_{s,t}\left\{{n + 2 \atop 2}\right\}_{s,t}.  (9)

Proof. We have

\left\{{n + 3 \atop 3}\right\}_{s,t} = \frac{\{n + 1\}_{s,t}\{n + 2\}_{s,t}\{n + 3\}_{s,t}}{\{3\}_{s,t}!}

= \frac{\{n + 1\}_{s,t}\{n + 2\}_{s,t}}{\{3\}_{s,t}!} (s\{n + 2\}_{s,t} + t\{n + 1\}_{s,t})

= \frac{\{n + 1\}_{s,t}\{n + 2\}_{s,t}}{\{3\}_{s,t}!} ((s^{2} + t)\{n + 1\}_{s,t} + st\{n\}_{s,t})

= \{n + 1\}_{s,t}\left\{{n + 2 \atop 2}\right\}_{s,t} + st\left\{{n + 2 \atop 3}\right\}_{s,t}.

The above theorem is Prop. 2.2 given by Sagan and Savage [1]. Eq. (9) is the Lucas analogue of the identity

Te_{n+1} + 2Te_{n} = (n + 1)T_{n+1}.

If we choose s = 1 + q and t = −q in Eq. (9), we obtain

\left[{n + 3 \atop 3}\right]_{q} = −(1 + q)q\left[{n + 2 \atop 3}\right]_{q} + \frac{1 − q^{n+1}}{1 − q} \left[{n + 2 \atop 2}\right]_{q}.

5 Generating functions

Theorem 8. For all n ≥ 1,

D_{φ,φ′}^{n}(\frac{1}{1 − x}) = \frac{\{n\}_{s,t}!}{(φ_{s,t}^{n}x; q)_{n+1}} ,  (10)

where (a; q)_{n} = ∏_{k=0}^{n−1}(1 − aq^{k}) is the q-shifted factorial.

Proof. Note that

D_{φ,φ′}(\frac{1}{1 − x}) = \frac{1}{(1 − φ_{s,t}x)(1 − φ′_{s,t}x)} = \frac{\{1\}_{s,t}!}{(φ_{s,t}x; q)_{2}} .

Suppose that Eq. (10) is true for n and let us prove by induction for n + 1. As

D_{φ,φ′}(φ_{s,t}^{n}x; q)_{n+1} = −{n + 1}_{s,t}(φ_{s,t}^{n}φ′_{s,t}x; q)_{n},

then

D_{φ,φ′}^{n+1}(\frac{1}{1 − x}) = D_{φ,φ′}D_{φ,φ′}^{n}(\frac{1}{1 − x})

= D_{φ,φ′}(\frac{\{n\}_{s,t}!}{(φ_{s,t}^{n}x; q)_{n+1}} )

= \frac{−\{n\}_{s,t}!D_{φ,φ′}(φ_{s,t}^{n}x; q)_{n+1}}{(φ_{s,t}^{n+1}x; q)_{n+1}(φ_{s,t}^{n}φ′_{s,t}x; q)_{n+1}}

= \frac{\{n + 1\}_{s,t}!(φ_{s,t}^{n}φ′_{s,t}x; q)_{n}}{(φ_{s,t}^{n+1}x; q)_{n+1}(φ_{s,t}^{n}φ′_{s,t}x; q)_{n+1}}

= \frac{\{n + 1\}_{s,t}!}{(1 − (φ_{s,t}q)^{n+1}x)(φ_{s,t}^{n+1}x; q)_{n+1}}

= \frac{\{n + 1\}_{s,t}!}{(φ_{s,t}^{n+1}x; q)_{n+2}} .

The proof is completed. Theorem 9. For all d ∈ ℕ,

∑_{n=1}^{∞} \left\{{n + d − 1 \atop d}\right\}_{s,t}x^{n} = \frac{x}{(φ_{s,t}^{d}x; q)_{d+1}} .

Proof. From Theorem 8,

\frac{x}{(φ_{s,t}^{d}x; q)_{d+1}} = \frac{x}{\{d\}_{s,t}!}D_{φ,φ′}^{d}(\frac{1}{1 − x})

= \frac{x}{\{d\}_{s,t}!}D_{φ,φ′}^{d}(∑_{n=0}^{∞} x^{n})

= \frac{x}{\{d\}_{s,t}!} ∑_{n=d}^{∞}\{n\}_{s,t}\{n − 1\}_{s,t} ⋯ \{n − d + 1\}_{s,t}x^{n−d}

= \frac{x}{\{d\}_{s,t}!} ∑_{n=0}^{∞} \frac{\{d + n\}_{s,t}!}{\{n\}_{s,t}!}x^{n}

= ∑_{n=1}^{∞} \left\{{n + d − 1 \atop d}\right\}_{s,t}x^{n}.

Theorem 10. The generating function of the squares of the Lucas-triangular numbers is

∑_{n=1}^{∞} \left\{{n + 1 \atop 2}\right\}_{s,t}^{2}x^{n}

= \frac{x + (\{4\}φ_{s,t} − φ_{s,t}^{4})x^{2} − \{3\}_{s,t}φ_{s,t}^{3}φ_{s,t}^{′3}x^{3} + (\{3\}_{s,t}φ_{s,t}^{7}φ_{s,t}^{′3} − \{4\}_{s,t}φ_{s,t}^{6}φ_{s,t}^{′3})x^{4}}{(1 − t^{2}x)(1 − φ_{s,t}^{′4}x)(φ_{s,t}^{4}x; q)_{4}} .

Proof. From Eq. (8),

∑_{n=1}^{∞} \left\{{n + 1 \atop 2}\right\}_{s,t}^{2}x^{n}

= ∑_{n=1}^{∞} ∑_{k=1}^{n} t^{2(n−k)}[k]_{φ^{2},φ^{′2}}[k]_{φ,φ′}^{2}x^{n}

= ∑_{k=1}^{∞}[k]_{φ^{2},φ^{′2}}[k]_{φ,φ′}^{2}x^{k} ∑_{n=k}^{∞} t^{2(n−k)}x^{n}

= ∑_{k=1}^{∞}[k]_{φ^{2},φ^{′2}}[k]_{φ,φ′}^{2}x^{k} ∑_{n=0}^{∞}(t^{2}x)^{n}

= \frac{1}{1 − t^{2}x}(xD_{φ,φ′})^{2}(xD_{φ^{2},φ^{′2}}) \{\frac{x}{1 − x}\}

= \frac{x + (\{4\}_{s,t}φ_{s,t} − φ_{s,t}^{4})x^{2} − \{3\}_{s,t}φ_{s,t}^{3}φ_{s,t}^{′3}x^{3} + (\{3\}_{s,t}φ_{s,t}^{7}φ_{s,t}^{′3} − \{4\}_{s,t}φ_{s,t}^{6}φ_{s,t}^{′3})x^{4}}{(1 − t^{2}x)(1 − φ_{s,t}^{′4}x)(φ_{s,t}^{4}x; q)_{4}} .

The q-analogue of the previous theorem is

∑_{n=1}^{∞} \left[{n + 1 \atop 2}\right]_{q}^{2}x^{n} = \frac{x + ([4]_{q} − 1)x^{2} − [3]_{q}q^{3}x^{3} + ([3]_{q}q^{3} − [4]_{q}q^{3})x^{4}}{(1 − q^{2}x)(1 − q^{4}x)(x; q)_{4}} .

6 Sum of reciprocals

The Lucas-zeta function, or (s, t)-zeta function, is the function defined by

ζ_{s,t}(z) = ∑_{n=1}^{∞} \frac{1}{\{n\}_{s,t}^{z}} = 1 + \frac{1}{s^{z}} + \frac{1}{(s^{2} + t)^{z}} + \frac{1}{(s^{3} + 2st)^{z}} + · · · .

Egami [4] and Navas [9] independently studied the zeta function ζ_{1,1}(z). Landau [8] studied the problem of evaluating ζ_{1,1}(1). The function ζ_{s,t}(z) is convergent when z > 0 and when either φ_{s,t} > 1 and 0 < |q| < 1 or φ′_{s,t} > 1 and |q| > 1. Take z = σ + iβ. Then ζ_{s,t}(z) converges when ℜ(z) > 0. Some specializations of ζ_{s,t}(1) are

ζ_{1,1}(1) = ζ_{F} (1) = ∑_{n=1}^{∞} \frac{1}{F_{n}} = 1 + 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{5} + \frac{1}{8} + ⋯ ≐ 3.359885666243 …,
ζ_{2,1}(1) = ζ_{P} (1) = ∑_{n=1}^{∞} \frac{1}{P_{n}} = 1 + \frac{1}{2} + \frac{1}{5} + \frac{1}{12} + \frac{1}{29} + ⋯ ≐ 1.81781609195402 …,
ζ_{1,2}(1) = ζ_{J}(1) = ∑_{n=1}^{∞} \frac{1}{J_{n}} = 1 + 1 + \frac{1}{3} + \frac{1}{5} + \frac{1}{11} + \frac{1}{21} + ⋯ ≐ 2.67186147 …,
ζ_{3,−2}(1) = ζ_{M}(1) = ∑_{n=1}^{∞} \frac{1}{M_{n}} = 1 + \frac{1}{3} + \frac{1}{7} + \frac{1}{15} + \frac{1}{31} + ⋯ ≐ 1.57511520737327 …,
ζ_{2t,−1}(1) = ζ_{U}(1) = ∑_{n=1}^{∞} \frac{1}{U_{n−1}(t)} = 1 + \frac{1}{2t} + \frac{1}{4t^{2} − 1} + \frac{1}{8t^{3} − 4t} + \frac{1}{16t^{4} − 12t^{2} + 1} + ⋯,

with t ≠ 0, cos \frac{kπ}{n+1}, and k = 1, 2, …, n. Theorem 11.

∑_{n=1}^{∞} \frac{t^{n}}{\{2n\}_{s,t}} = \sqrt{s^{2} + 4t}(L(φ_{s,t}^{′2}/t) − L(φ_{s,t}^{′4}/t^{2})),

where L(q) = ∑_{n=1}^{∞} \frac{q^{n}}{1−q^{n}} is the Lambert function. Proof. Take into account that

\frac{t^{n}}{\{2n\}_{s,t}} = \sqrt{s^{2} + 4t}(\frac{(φ_{s,t}^{′2}/t)^{n}}{1 − (φ_{s,t}^{′2}/t)^{n}} − \frac{(φ_{s,t}^{′4}/t^{2})^{n}}{1 − (φ_{s,t}^{′4}/t^{2})^{n}} )

and sum for all n ≥ 1. Theorem 12.

∑_{n=1}^{∞} \frac{1}{\{2n − 1\}_{s,1}} = −\frac{\sqrt{s^{2} + 4}}{4}θ_{2}(φ_{s,1}^{′2})^{2},

where θ_{2}(q) = ∑_{n=−∞}^{∞} q^{(n+1/2)^{2}}. Proof.

∑_{n=1}^{∞} \frac{1}{\{2n − 1\}_{s,1}} = −\frac{\sqrt{s^{2} + 4}}{2} ∑_{n=1}^{∞} \frac{2(φ_{s,1}^{′2})^{n−1/2}}{1 + (φ_{s,1}^{′2})^{2n−1}}

= −\frac{\sqrt{s^{2} + 4}}{4} ∑_{a=−∞}^{∞} \frac{2(φ_{s,1}^{′2})^{a−1/2}}{1 + (φ_{s,1}^{′2})^{2a−1}}

= −\frac{\sqrt{s^{2} + 4}}{4}θ_{2}(φ_{s,1}^{′2})^{2}.

Theorem 13.

∑_{n=1}^{∞} \frac{1}{\{n\}_{s,t}\{n + 1\}_{s,t}} = φ_{s,t} − (1 + t) ln_{s,t}(1 + \frac{φ_{s,t}}{t} ),  (11)

where

ln_{s,t}(1 − x) = − ∑_{n=1}^{∞} \frac{x^{n}}{\{n\}_{s,t}}

is a Lucas analogue of the logarithm function.

Proof. Suppose that 0 < |q| < 1, and q = φ′_{s,t}/φ_{s,t}, and set a_{n} = 1/{n}_{s,t}{n + 1}_{s,t}. As

lim_{n→∞} \frac{a_{n+1}}{a_{n}} = lim_{n→∞} \frac{\{n\}_{s,t}}{\{n + 2\}_{s,t}} = lim_{n→∞} \frac{1 − q^{n}}{φ_{s,t}^{2} − φ_{s,t}^{′2}q^{n}} = \frac{1}{φ_{s,t}^{2}} ,

then the series in the left-hand of Eq. (11) is convergent only if |φ_{s,t}| > 1. If |q| > 1, then the series in the left-hand of Eq. (11) is convergent only if |φ′_{s,t}| > 1. By partial fractions

\frac{1}{\{n\}_{s,t}\{n + 1\}_{s,t}} = \frac{A}{\{n\}_{s,t}} + \frac{B}{\{n + 1\}_{s,t}} = \frac{A}{\{n\}_{s,t}} + \frac{B}{φ_{s,t}\{n\}_{s,t} + φ_{s,t}^{′n}} ,

where A = (−φ_{s,t}/t)^{n} and B = (−φ_{s,t}/t)^{n+1}. Then

∑_{n=1}^{∞} \frac{1}{\{n\}_{s,t}\{n + 1\}_{s,t}} = ∑_{n=1}^{∞}(\frac{(−φ_{s,t}/t)^{n}}{\{n\}_{s,t}} + \frac{t(−φ_{s,t}/t)^{n+1}}{\{n + 1\}_{s,t}} )

= −\frac{φ_{s,t}}{t} + (1 + t) ∑_{n=2}^{∞} \frac{(−φ_{s,t}/t)^{n}}{\{n\}_{s,t}}

= −\frac{φ_{s,t}}{t} − (1 + t)(ln_{s,t}(1 + \frac{φ_{s,t}}{t} ) − \frac{φ_{s,t}}{t} )

= φ_{s,t} − (1 + t) ln_{s,t}(1 + \frac{φ_{s,t}}{t} ).

The function ln_{s,t}(1 − x) is convergent for all x ∈ (−|φ_{s,t}|, |φ_{s,t}|).

Some specializations of Theorem 13 are

∑_{n=1}^{∞} \frac{1}{F_{n}F_{n+1}} = \frac{1 + \sqrt{5}}{2} − 2 ln_{F} (\frac{3 + \sqrt{5}}{2}),

∑_{n=1}^{∞} \frac{1}{P_{n}P_{n+1}} = 1 + \sqrt{2} − 2 ln_{P} (2 + \sqrt{2}),

∑_{n=1}^{∞} \frac{1}{J_{n}J_{n+1}} = 2 − 3 ln_{J}(2),

∑_{n=1}^{∞} \frac{1}{M_{n}M_{n+1}} = 2 + ln_{M}(0),

where ln_{F} ≡ ln_{1,1}, ln_{P} ≡ ln_{2,1}, ln_{J} ≡ ln_{1,2}, and ln_{M} ≡ ln_{3,−2}. A very important remark is that ln_{M}(0) is finite.

Corollary 14.

∑_{n=1}^{∞} \frac{(−t)^{n}}{\{n\}_{s,t}\{n + 1\}_{s,t}} = \frac{s + \sqrt{s^{2} + 4t}}{2}.

Proof. If t = −1 in the previous theorem, then

∑_{n=1}^{∞} \frac{1}{\{n\}_{s,−1}\{n + 1\}_{s,−1}} = φ_{s,−1}.

Note that \{n\}_{s,−1} = (i/\sqrt{t})^{n−1}\{n\}_{a,t}, where s = ia/\sqrt{t}. Then

∑_{n=1}^{∞} \frac{1}{\{n\}_{s,−1}\{n + 1\}_{s,−1}} = ∑_{n=1}^{∞} \frac{1}{(i/\sqrt{t})^{2n−1}\{n\}_{a,t}\{n + 1\}_{a,t}}

= \frac{i}{\sqrt{t}} ∑_{n=1}^{∞} \frac{(−t)^{n}}{\{n\}_{a,t}\{n + 1\}_{a,t}} = \frac{i}{\sqrt{t}}φ_{a,t}.

The proof is completed.

Some specializations of Corollary 14 are

∑_{n=1}^{∞} \frac{(−1)^{n}}{F_{n}F_{n+1}} = \frac{1 + \sqrt{5}}{2}.

∑_{n=1}^{∞} \frac{(−1)^{n}}{P_{n}P_{n+1}} = 1 + \sqrt{2}.

∑_{n=1}^{∞} \frac{(−2)^{n}}{J_{n}J_{n+1}} = 2.

∑_{n=1}^{∞} \frac{2^{n}}{M_{n}M_{n+1}} = 2.

∑_{n=1}^{∞} \frac{1}{U_{n−1}(t)U_{n}(t)} = t + \sqrt{t^{2} − 1}.

References

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[2] R. B. Corcino, On P, Q-binomial coefficients, Integers 8 (2008), Paper #A29.

[3] E. Deza and M. M. Deza, Figurate Numbers, World Scientific, 2012.

[4] S. Egami, Some curious Dirichlet series, RIMS Kokyuroku 1091 (1999), 172–174.

[5] S. Ekhad, The Sagan-Savage Lucas-Catalan polynomials have positive coefficients, arxiv preprint arXiv:1101.4060 [math.CO], 2011. Available at http://arxiv.org/abs/1101.

4060.

[6] G. Gasper and M. Rahman, Basic Hypergeometric Series, 2^{nd} ed., Cambridge University Press, 1990.

[7] S. Kus, N. Tuglu, and T. Kim, Bernoulli F -polynomials and Fibo-Bernoulli matrices, Adv. Differential Equations 145 (2019).

[8] E. Landau, Sur la série des inverse de nombres de Fibonacci, Bull. Soc. Math. France 27 (1899), 298–300.

[9] L. Navas, Analytic continuation of the Fibonacci Dirichlet series, Fibonacci Quart. 39 (2001), 409–418.

[10] J. Ruiz, A Lucas analogue of Eulerian numbers, arxiv preprint arXiv:2301.05215

[math.CO], 2023. Available at http://arxiv.org/abs/2301.05215.

[11] B. Sagan and C. Savage, Combinatorial interpretations of binomial coefficient analogues related to Lucas sequences, Integers 10 (2010), #A52.

[12] M. J. Schlosser, q-analogues of the sums of consecutive integers, squares, cubes, quarts and quints, Electron. J. Comb. 11 (2004), #R71.

[13] N. J. A. Sloane et al., The On-Line Encyclopedia of Integer Sequences, 2025. Available

at https://oeis.org/.

[14] S. O. Warnaar, On the q-analogue of the sum of cubes, Electron. J. Comb. 11 (2004), #N13.

2020 Mathematics Subject Classification: Primary 11B39; Secondary 05A15, 11B65.

Keywords: simplicial d-polytopic number, generalized Fibonacci polynomial, generalized triangular number, generalized tetrahedral number.

(Concerned with sequences \underline{A001654}, \underline{A084158}, \underline{A084175}, \underline{A006095}, \underline{A001655}, \underline{A099930}, and \underline{A006096}.)

Received March 18 2025; revised version received August 13 2025; September 20 2025. Published in Journal of Integer Sequences, September 20 2025.

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