The Golden Ratio, Factorials, and the Lambert W Function

Aleksandar Petojević, Marjana Gorjanac Ranitović, Dragan Rastovac, and

Milinko Mandić University in Novi Sad

Faculty of Education in Sombor Podgorička 4 25000 Sombor Serbia

 apetoje@uns.ac.rs
ranitovicm@uns.ac.rs
rastovacd@gmail.com
milinmand@gmail.com

Abstract

We study the relationship of the integer sequence \underline{A214048} with the Lambert W function and the left factorial numbers \underline{A003422}.

1 Notation and Introduction

We use the following notation [7]:

• ℕ: the set of positive integers.

• ℕ_{0}: the set of non-negative integers.

• ℤ_{0}^{−}: the set of negative integers and zero.

• ℝ: the set of real numbers.

• ℝ^{+}: the set of real positive numbers.

• ℂ: the set of complex numbers.

• L_{m}: the m’th Lucas number [6, \underline{A000032}].

• F_{m}: the m’th Fibonacci number [6, \underline{A000045}].

• AM_{m}: the m’th associated Mersenne number [6, \underline{A001350}].

• φ: the golden ratio, φ = \frac{1+\sqrt{5}}{2}.

• ⌊·⌋: the floor function of a real number, which is the largest integer not exceeding that real number.

• W (·): the Lambert W function [4] is the solution to the equation xe^{x} = z for z ∈ ℂ.

• Re(·): the real part of the complex number.

• Γ(·): the gamma function, defined by [8, p. 1]

Γ(z) = ∫_{0}^{+∞} e^{−t}t^{z−1}dt (Re(z) > 0).

• ψ^{(m)}(·): the polygamma function of order m, [8, p. 22], defined by

ψ^{(m)}(z) = \frac{∂^{m}}{∂z^{m}} ψ(z) = \frac{∂^{m+1}}{∂z^{m+1}} ln(Γ(z)),
ψ^{(0)}(z) = ψ(z) = \frac{Γ′(z)}{Γ(z)}  (m ∈ ℕ_{0}, z ∈ ℂ\ℤ_{0}^{−}).

• arg(·): the argument of the complex number.

• γ(·, ·): the incomplete gamma function, [8, p. 11], defined by γ(s, a) = ∫_{0}^{a} e^{−t}t^{s−1}dt (Re(s) > 0, | arg(a)| < π).

Next, let us define the function

!z(x) = ∫_{0}^{x} e^{−t} \frac{t^{z} − 1}{t − 1}dt (x ∈ ℝ, z ∈ ℂ).

On the basis of known properties of the Lambert W function and the sequence [6, \underline{A214048}], we determine x ∈ ℝ so that

!n(x) < !n < !n(x) + 1 (n ∈ ℕ).

2 Connection to the Lambert W function

As a multi-valued function, W (z) has infinitely many complex branches and two real branches [3, p. 2]:

W_{0} : [−1/e, ∞) → [−1, ∞),
W_{−1} : [−1/e, 0) → (−∞, −1].

In what follows, we consider only the real branch W_{−1}(z) of the Lambert W function. The function W_{−1}(z) strictly decreases on (−1/e, 0) and W_{−1}(−1/e) = −1. We also have the following well-known inequality [3, p. 3] for the function W_{−1}(z):

\frac{e ln(−x)}{e − 1} ≤ W_{−1}(x) ≤ ln(−x) − ln(− ln(−x)) (x ∈ [−1/e, 0)).  (1)

In proving one of the main results of the paper, the polygamma function of order m plays a key role. The polygamma function satisfies the inequality

\frac{(m − 1)!}{x^{m}} + \frac{m!}{2x^{m+1}} ≤ (−1)^{(m+1)}ψ^{(m)}(x) ≤ \frac{(m − 1)!}{x^{m}} + \frac{m!}{x^{m+1}} (m ≥ 1, x > 0). (2)

2.1 The real sequence (a_{n})_{n≥4}

We have −\frac{1}{n−1}(4/7)^{\frac{1}{n−1}} ∈ (−\frac{1}{e}, 0), (4 ≤ n ∈ ℕ). Hence the equation xe^{x} = −\frac{1}{n−1}(4/7)^{\frac{1}{n−1}} has two real solutions, namely x = W_{0}(−\frac{1}{n−1}(4/7)^{\frac{1}{n−1}} ) and x = W_{−1}(−\frac{1}{n−1}(4/7)^{\frac{1}{n−1}} ). Let us define the sequence of real positive numbers (a_{n})_{n≥4} as follows:

a_{n} = −(n − 1)W_{−1}(−\frac{1}{n − 1}(4/7)^{\frac{1}{n−1}} ) (a_{n} ∈ ℝ^{+}).  (3)

Remark 1. Note that the sequence a_{n} is strictly increasing. We have

a_{n}^{−(n−1)}e^{a_{n}} = a_{n+1}^{−n}e^{a_{n+1}} and a_{n+1} = a_{n} + ln(a_{n+1}(\frac{a_{n+1}}{a_{n}} )^{n−1}).

2.2 The golden sequence (α_{n})_{n≥1}

In 2012, Kimberling [6] defined the golden sequence (α_{n})_{n≥1} [6, \underline{A214048}] by

α_{n} = m ⟺ φ^{m−1} ≤ n! ≤ φ^{m} (m ∈ ℕ),  (4)

or

α_{1} = 1, α_{n} = m ⟺ L_{m−1} ≤ n! < Ł_{m} (1 < n ∈ ℕ).

Definition (4) produces the following lemma:

Lemma 2. The following inequalities hold:

α_{n−1} + ⌊\frac{ln n}{ln φ}⌋ ≤ α_{n} ≤ α_{n−1} + ⌊\frac{ln n}{ln φ}⌋ + 1 (1 < n ∈ ℕ).

Consider the numbers ⌊\frac{ln n}{ln φ}⌋ [6, \underline{A371672}]. For this purpose, we consider the integer sequence (d_{m})_{m≥0} [6, \underline{A181716}] (see also [6, \underline{A098600}]) which we define as follows:

d_{m} = d_{m−1} + d_{m−2} + (−1)^{m}, d_{0} = 0, d_{1} = 1.

The following equalities hold [6]:

d_{m} = F_{m−2} + F_{m} + (−1)^{m} = AM_{m−1} +1 (1 < m ∈ ℕ).

Lemma 3. The number d_{m} is the number of natural numbers n for which the equality

⌊\frac{ln n}{ln φ}⌋ = m (m ∈ ℕ)

holds.

Proof. The proof presented here is due to J. Shallit. If we want to count the number of n for which ⌊\frac{ln n}{ln φ}⌋ = m, this is the same as counting the number of integers n such that m ≤ \frac{ln n}{ln φ} < m + 1, which by rearrangement is the number of integers n such that φ^{m} ≤ n < φ^{m+1}. However, we know from the Binet form of the Lucas numbers that

⌊φ^{m}⌋ = \begin{cases}L_{m} − 1, & if m is even; \\ L_{m}, & if m is odd, \end{cases}

so now trivially we get d_{m} = L_{m−1} + 1 if m is even and d_{m} = L_{m−1} − 1 if m is odd, that is,

d_{m} = L_{m−1} + (−1)^{m} (m ∈ ℕ).  (5)

\begin{matrix}n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 \\ ⌊\frac{ln n}{ln φ}⌋ & 0 & 1 & 2 & 2 & 3 & 3 & 4 & 4 & 4 & 4 & 4 & 5 & 5 \end{matrix}

Table 1: The numbers ⌊\frac{ln n}{ln φ}⌋ for n = 1, 2, …, 13.

Remark 4. Equality (5) was first stated without proof by G. C. Greubel in [6, \underline{A181716}]. We do not use Lemma 2 and Lemma 3 when proving the new results.

Example 5. We have

⌊\frac{ln n}{ln φ}⌋ = 1 for n = 2  ⇒ d_{1} = 1,
⌊\frac{ln n}{ln φ}⌋ = 2 for n = 3, 4  ⇒ d_{2} = 2,
⌊\frac{ln n}{ln φ}⌋ = 3 for n = 5, 6  ⇒ d_{3} = 2,
⌊\frac{ln n}{ln φ}⌋ = 4 for n = 7, 8, 9, 10, 11 ⇒ d_{4} = 5.

2.3 The main result

We give our main result in the following theorem:

Theorem 6. The following inequality holds:

α_{n} > a_{n} (4 < n ∈ ℕ).

Proof. The inequality is easily checked for 4 ≤ n ≤ 78. We prove that it holds for n > 78. Since φ^{α_{n}} > n!, we have α_{n} > \frac{ln n!}{ln φ} > 2 ln(n!). Inequality (1) produces

−(n − 1)\frac{e}{e − 1} ln(\frac{1}{n − 1}(4/7)^{\frac{1}{n−1}} ) ≥ a_{n}.

Therefore we have to prove 2 ln(n!) > a_{n}, or, equivalently

2 ln(n!) > −(n − 1)\frac{e}{e − 1} ln(\frac{1}{n − 1}(4/7)^{\frac{1}{n−1}} ).  (6)

The last inequality holds for n = 79. Let us define the function

ω(x) = 2 ln(Γ(x + 1)) + (x − 1)\frac{e}{e − 1} ln(\frac{1}{x − 1}(4/7)^{\frac{1}{x−1}} ).

Therefore we need to prove that the first derivative of the function ω(x) is higher than 0 for 79 ≤ x ∈ ℝ. It is easy to check that

ω′(x) = 2ψ(x + 1) + \frac{e}{e − 1}(ln(\frac{1}{x − 1}(4/7)^{\frac{1}{x−1}} ) − 1 + \frac{ln 7/4}{x − 1}) > 0 (x = 38).

Further, it is necessary to show that ω′(x) is an increasing function for 79 ≤ x ∈ ℝ. We need to prove the following inequality:

\frac{∂^{2}}{∂x^{2}} ω(x) = 2ψ^{(1)}(x + 1) − \frac{e}{(e − 1)(x − 1)} > 0 (79 ≤ x ∈ ℝ).

Hence using Inequality (2) it is sufficient to show that

2(\frac{1}{x + 1} + \frac{1}{2(x + 1)^{2}} ) > \frac{e}{(e − 1)(x − 1)}.

Equivalently, it suffices to show that

2x^{2} + x − 3 > (x + 1)^{2} \frac{16}{10} > (x + 1)^{2} \frac{e}{(e − 1)}.

It follows straightforwardly that

2x^{2} + x − 3 > (x + 1)^{2} \frac{16}{10}

holds for x ≥ 8, which completes our proof.

3 The left factorial

In 1971, Kurepa [1] defined the function left factorial [6, \underline{A003422}] for natural numbers !n by

!0 = 0, !n = ∑_{k=0}^{n−1} k! (n ∈ ℕ).

In the aforementioned paper [1, p. 151] Kurepa extended left factorial function to the complex half-plane Re(z) > 0 as

!z = ∫_{0}^{+∞} e^{−t} \frac{t^{z} − 1}{t − 1}dt.

We can also extend such function analytically to the whole complex plane [2] by

!z = !(z + 1) − Γ(z + 1).

Remark 7. In [1, p. 149], Kurepa proposed the conjecture for the left factorial as follows:

If 1 < n ∈ ℕ, then gcd(!n, n!) = 2.

Over the past fifty years there have been many attempts to find a solution to Kurepa’s conjecture. The problem remains open. For more details, see [5].

3.1 The left factorial and the Lambert W function

Let us define the sequence of real positive numbers (b_{n})_{n≥4} as follows:

b_{n} = ∫_{0}^{a_{n}} e^{−x} \frac{x^{n} − 1}{x − 1}dx = ∑_{k=0}^{n−1} γ(k + 1, a_{n}) (b_{n} ∈ ℝ^{+}).

The sequence a_{n} is given by (3).

\begin{matrix}n & 4 & 5 & 6 & 7 & 8 & 9 \\ a_{n} & 5.87 & 9.61 & 13.62 & 17.85 & 22.29 & 26.90 \\ b_{n} & 8.86 & 33.01 & 153.08 & 873.12 & 5913.15 & 46233.17 \\ !n & 10 & 34 & 154 & 874 & 5914 & 46234 \end{matrix}

Table 2: The numbers a_{n}, b_{n} and !n for n = 4, 5, 6, 7, 8, 9.

Lemma 8. We have

2n − 4 < \frac{3}{4}a_{n} (4 ≤ n ∈ ℕ).

Proof. Applying Remark 1, we have

a_{n+1} > a_{n} + ln a_{n+1}  (4 ≤ n ∈ ℕ)

> a_{n} + 2.8 > a_{n} + \frac{8}{3} (ln a_{6+1} > 2.88).

Hence by induction on n we obtain our inequality. Lemma 9. The following inequality holds:

∑_{t=0}^{k} \frac{a_{n}^{t}}{t!} < \frac{a_{n}^{k+1}}{(k + 1)!} (5 ≤ n ∈ ℕ, n − 2 > k ∈ ℕ_{0}).

Proof. We prove the lemma by induction on k. For k = 0 we have 1 < a_{n}, which is valid. Assume that the statement holds for k − 1 < n − 3, i.e.,

∑_{t=0}^{k−1} \frac{a_{n}^{t}}{t!} < \frac{a_{n}^{k}}{k!} .

Then

∑_{t=0}^{k} \frac{a_{n}^{t}}{t!} = \frac{a_{n}^{k}}{k!} + ∑_{t=0}^{k−1} \frac{a_{n}^{t}}{t!} < 2\frac{a_{n}^{k}}{k!} .

Hence Lemma 8 produces

2\frac{a_{n}^{k}}{k!} < \frac{a_{n}^{k+1}}{(k + 1)!} (k < n − 2),

which completes the proof. Theorem 10. The following inequalities hold:

b_{n} < !n < b_{n} + 1 (5 ≤ n ∈ ℕ).

Proof. The claim holds for 5 ≤ n ≤ 9. Since

e^{−x} \frac{x^{n} − 1}{x − 1} > 0,

the left side of the inequality holds. Furthermore, we have

∫_{0}^{a_{n}} e^{−x} \frac{x^{n} − 1}{x − 1}dx = !n − e^{−a_{n}} ∑_{t=0}^{n−1} \frac{!n−!t}{t!}a_{n}^{t}.  (7)

To prove the right side of the inequality, we use Equation (7). It suffices to show

∑_{t=0}^{n−1} \frac{!n−!t}{t!}a_{n}^{t} < e^{a_{n}}.

Firstly, by induction on n we have the proof of the following inequality:

2n + 1 < \frac{3}{4} a_{n} (9 ≤ n ∈ ℕ).  (8)

Next, we have

∑_{t=0}^{n−1} \frac{!n−!t}{t!}a_{n}^{t} = a_{n}^{n−1} + na_{n}^{n−2} + ∑_{t=0}^{n−3} \frac{!n−!t}{t!}a_{n}^{t} < a_{n}^{n−1} + na_{n}^{n−2}+ !n ∑_{t=0}^{n−3} \frac{a_{n}^{t}}{t!}

< a_{n}^{n−1} + na_{n}^{n−2}+ !n\frac{a_{n}^{n−2}}{(n − 2)!} (Lemma 9 for k = n − 3)

< a_{n}^{n−1} + na_{n}^{n−2} + (n + 1)a_{n}^{n−2} (\frac{!(n − 2)}{(n − 2)!} < 1)

< \frac{7}{4}a_{n}^{n−1}. (Equation (8)).

Since one real solution to the equation

e^{x} − \frac{7}{4}x^{n−1} = 0

is a_{n}, we have

∑_{t=0}^{n−1} \frac{!n−!t}{t!}a_{n}^{t} < \frac{7}{4}a_{n}^{n−1} = e^{a_{n}},

which proves our theorem.

Remark 11. If we replace the constant \frac{4}{7} with numbers \frac{4+ \frac{1}{31}}{7} or \frac{4− \frac{1}{31}}{7} in Definition (3), then Lemma 9 and Inequality (8) hold for the sequence (a_{n})_{n≥4} defined in this way. We conclude that our constant \frac{4}{7} is not optimal. We do not deal with this issue in this paper.

3.2 The left factorial and the golden sequence

Let us define the sequence of real numbers (β_{n})_{n≥1} as follows:

β_{n} = ∫_{0}^{α_{n}} e^{−x} \frac{x^{n} − 1}{x − 1}dx (β_{n} ∈ ℝ).

The sequence α_{n} is given by (4).

\begin{matrix}n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ α_{n} & 1 & 2 & 4 & 7 & 10 & 14 & 18 & 23 \\ β_{n} & 0.63 & 1.46 & 3.13 & 9.44 & 33.23 & 153.29 & 873.21 & 5913.49 \\ !n & 1 & 2 & 4 & 10 & 34 & 154 & 874 & 5914 \end{matrix}

Table 3: The numbers !n, α_{n} and β_{n} for n = 1, 2, …, 8.

Theorem 12. The following inequalities hold:

β_{n} < !n < β_{n} + 1 (n ∈ ℕ).

Proof. Let us define the family of functions f_{n}(x) as follows:

f_{n}(x) = x − (n − 1) ln x − ln \frac{7}{4} (0 < x ∈ ℝ, 3 < n ∈ ℕ).

Then

\frac{∂}{∂x}f_{n}(x) = f′_{n}(x) = 1 − \frac{n − 1}{x}, \frac{∂^{2}}{∂x^{2}} f_{n}(x) = \frac{n − 1}{x^{2}}, f′_{n}(n − 1) = 0, f_{n}(n − 1) < 0.

For the equation f_{n}(x) = 0, we have the real solution x = a_{n}. On the interval (0, +∞) the function f_{n}(x) has the minimum for x = n − 1. The function f_{n}(x) increases on the interval (0, +∞). For the sequence (α_{n})_{n≥5} the analogs of Lemma 9 and Inequality (8) are valid. Hence Theorem 6 produces

β_{n} < !n < β_{n} + 1.

Remark 13. If Lemma 9 and Inequality (8) hold for the positive real sequence (c_{n})_{n≥n_{0}}, and if c_{n} ≥ a_{n}, (n ≥ n_{0} ≥ 5), then

∫_{0}^{c_{n}} e^{−x} \frac{x^{n} − 1}{x − 1}dx < !n < ∫_{0}^{c_{n}} e^{−x} \frac{x^{n} − 1}{x − 1}dx + 1 (n ≥ n_{0}).

3.3 Generalization

Let us define two-dimensional real sequences (A_{n,m})_{n≥5,m≥1} and (B_{n,m})_{n≥5,m≥1} as follows:

A_{n,m} = −(n − 1)W_{−1}(−\frac{1}{n − 1} (\frac{4}{7m})^{\frac{1}{n−1}} ),
B_{n,m} = ∫_{0}^{A_{n,m}} e^{−x} \frac{x^{n} − 1}{x − 1}dx.

\begin{matrix}n & 5 & 6 & 7 & 8 & 9 \\ B_{n,1} & 33.01 & 153.08 & 873.12 & 5913.15 & 46233.17 \\ B_{n,2} & 33.53 & 153.56 & 873.57 & 5913.58 & 46233.60 \\ B_{n,3} & 33.70 & 153.71 & 873.72 & 5913.73 & 46233.70 \\ B_{n,4} & 33.78 & 153.79 & 873.79 & 5913.80 & 46233.80 \\ B_{n,5} & 33.82 & 153.83 & 873.83 & 5913.84 & 46233.84 \\ !n & 34 & 154 & 874 & 5914 & 46234 \end{matrix}

Table 4: The numbers B_{n,m} and !n for n = 5, 6, 7, 8, 9 and m = 1, 2, 3, 4, 5.

Theorem 14. The following inequalities hold:

B_{n,m} < !n < B_{n,m} + \frac{1}{m} (5 ≤ n ∈ ℕ, m ∈ ℕ).

Proof. The Theorem 14 follows from the application of the procedures used in the proof of Theorem 10 and Remark 13. Corollary 15. The following equalities hold:

!n = ⌊β_{n}⌋ + 1  (n ∈ ℕ)

= ⌊b_{n}⌋ + 1 = ⌊B_{n,m} + \frac{1}{m}⌋ (4 < n ∈ ℕ).

4 Acknowledgment

We would like to thank the referee for his helpful suggestions.

References

[1] -D. Kurepa, On the left factorial function !n, Math. Balkanica 1 (1971), 147–153.

[2] -D. Kurepa, Left factorial function in complex domain, Math. Balkanica 3 (1973), 297–307.

[3] L. Lóczi, Guaranteed and high precision evaluation of the Lambert W function, Appl. Math. Comput. 433 (2022), Article 127406.

[4] I. Mezö, The Lambert W Function, Its Generalisations and Applications, Chapman and Hall/CRC, 2022.

[5] A. Petojević, S. Gordić, M. Mandić, and M. Gorjanac Ranitović, New equivalents of Kurepa’s hypothesis for left factorial, Axioms 12 (8) (2023), Article 785.

[6] N. J. A. Sloane et al., The On-Line Encyclopedia of Integer Sequences, 2024. Available

   at https://oeis.org.

[7] S. T. Somu and D. V. K. Tran, On sums of practical numbers and polygonal numbers, J. Integer Sequences 27 (2024), Article 24.5.1.

[8] H. M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, 2001.

2020 Mathematics Subject Classification: Primary 11B83; Secondary 11B39.

Keywords: golden ratio, Fibonacci number, Lucas number, associated Mersenne number, Lambert W function, left factorial, Kurepa hypothesis.

(Concerned with sequences \underline{A000032}, \underline{A000045}, \underline{A001350}, \underline{A003422}, \underline{A098600}, \underline{A181716}, \underline{A214048} and \underline{A371672}.)

Received April 4 2024; revised versions received April 5 2024; May 4 2024; May 5 2024; May 7 2024; May 15 2024. Published in Journal of Integer Sequences, June 3 2024.

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