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Extended Bowser sum

Robert Reynolds

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Distributed under a Creative Commons Attribution 4.0 International License
# Extended Bowser sum

Robert Reynolds

**ABSTRACT.** A finite sum of the Hurwitz-Lerch Zeta function $\Phi(z, s, a)$ is derived in terms of the Hurwitz-Lerch Zeta function. As a by-product of this investigation finite products of trigonometric functions in terms of the product of trigonometric functions, the gamma function and a recurrence identities are obtained. A special case of involving the finite product of the gamma function in terms of the ratio of the gamma function is produced. This form was previously derived by Gauss, Nielsen and Knar.

## 1. Introduction

Although numerous finite products and sums containing trigonometric functions have been collected and published by Hansen [1] and Bowser [2] along with extensive literature in [3] many significant gaps still remain. By browsing scholarly databases or reviewing textbooks and academic publications in the area of number theory, one may find specific research articles and academic publications on the finite sum of the Hurwitz-Lerch zeta function in the context of composite function evaluations and derivations using contour integration.

In this present work we derive a new expression for the Hurwitz-Lerch Zeta function $\Phi(z, s, a)$ in terms of the sum of two Hurwitz-Lerch zeta functions given by

$$ (1.1) \qquad \begin{aligned} & -\sum_{p=0}^{n-1} i2^{k+1} (i2^p)^k e^{im2^{p+1}} \left( \Phi(-e^{i2^{p+1}m}, -k, 1 - i2^{-p-1}\log(a)) \right. \\ & \qquad \left. \qquad + \Phi\left(e^{i2^{p+1}m}, -k, 1 - i2^{-p-1}\log(a)\right) \right) \\ & = i2^{k+2} \left( (i2^n)^k e^{im2^{n+1}} \Phi\left(e^{i2^{n+1}m}, -k, 1 - i2^{-n-1}\log(a)\right) \right. \\ & \qquad \left. \qquad -i^k e^{2im} \Phi\left(e^{2im}, -k, 1 - \frac{1}{2}i\log(a)\right) \right) \end{aligned} $$

<footnote n="2020">2020 Mathematics Subject Classification. Primary 30E20, 33-01, 33-03, 33-04.</footnote>

<footnote>Key words and phrases. Trigonometric functions, Cauchy integral, Hurwitz-Lerch Zeta function, gamma function, Nielsen form, Gauss's formula.</footnote>
where the variables $k, a, m$ are general complex numbers and $n$ is any prime number. The derivations follow the method used by us in [4]. This method involves using a form of the generalized Cauchy's integral formula given by

$$ (1.2) \qquad \frac{y^k}{\Gamma(k+1)} = \frac{1}{2\pi i} \int_C \frac{e^{wy}}{w^{k+1}} dw, $$

where $y, w \in \mathbb{C}$ and $C$ is in general an open contour in the complex plane where the bilinear concomitant [4] has the same value at the end points of the contour. This method involves using a form of equation (1.2) then multiplies both sides by a function, then takes the finite sum of both sides. This yields a finite sum in terms of a contour integral. Then we multiply both sides of equation (1.2) by another function and take the infinite sum of both sides such that the contour integral of both equations are the same.

## 2. The Hurwitz-Lerch Zeta function

We use equation (1.11.3) in [5] where $\Phi(z, s, v)$ is the Lerch function which is a generalization of the Hurwitz zeta $\zeta(s, v)$ and Polylogarithm functions $Li_n(z)$. The Lerch function has a series representation given by

$$ (2.1) \qquad \Phi(z, s, v) = \sum_{n=0}^{\infty} (v+n)^{-s} z^n $$

where $|z|<1, v \neq 0, -1, -2, -3, \dots$, and is continued analytically by its integral representation given by

$$ (2.2) \qquad \Phi(z, s, v) = \frac{1}{\Gamma(s)} \int_{0}^{\infty} \frac{t^{s-1} e^{-(v-1)t}}{e^t - z} dt $$

where $\mathrm{Re}(v) > 0$, and either $|z| \le 1, z \ne 1, \mathrm{Re}(s) > 0$, or $z = 1, \mathrm{Re}(s) > 1$.

## 3. Finite Sum of the Contour Integrals

### 3.1. Derivation of the first contour integral.
We use the method in [4]. The cut and contour are in the first quadrant of the complex $w$-plane with $0 < \mathrm{Re}(w+m) < 1$. The cut approaches the origin from the interior of the first quadrant and goes to infinity vertically and the contour goes round the origin with zero radius and is on opposite sides of the cut. Using a generalization of Cauchy's integral formula (1.2) we first replace $y \to \log(a) + i2^{p+1}(y+1)$ then multiply both sides by $-2ie^{im2^{p+1}(y+1)}$ and take the infinite and finite sums respectively over $y \in [0, \infty)$ and $p \in [0, n-1]$ and simplify in terms of the Hurwitz-Lerch Zeta function to get

$$ (3.1) \qquad -\sum_{p=0}^{n-1} \frac{i2^{k+1} (i2^p)^k e^{im2^{p+1}} \Phi\left(e^{i2^{p+1}m}, -k, 1 - i2^{-p-1}\log(a)\right)}{\Gamma(k+1)} \\ = -\frac{1}{2\pi i} \sum_{y=0}^{\infty} \sum_{p=0}^{n-1} \int_C 2ia^w w^{-k-1} e^{i2^{p+1}(y+1)(m+w)} dw $$
$$ = -\frac{1}{2\pi i} \int_C \sum_{p=0}^{n-1} \sum_{y=0}^{\infty} 2ia^w w^{-k-1} e^{i2^{p+1}(y+1)(m+w)} dw \\ = \frac{1}{2\pi i} \int_C \sum_{p=0}^{n-1} (a^w w^{-k-1} \cot(2^p(m+w)) + ia^w w^{-k-1}) dw $$

from equation (1.232.1) in [6] where $Im(m + w) > 0$, $Re(m + w) > 0$, $n$ is a prime number in order for the sum to converge. We apply Tonelli's theorem for multiple sums, see page 189 in [7] as the summand is of bounded measure over the space $\mathbb{C} \times [0, n-1] \times [0, \infty)$.

**3.2. The Additional Contour Integral.** Using a generalization of Cauchy's integral formula (1.2) we first replace $y \to \log(a)$ and multiply both sides by $i$ simplify to get

$$ (3.2) \qquad \sum_{p=0}^{n-1} \frac{i \log^k(a)}{\Gamma(k+1)} = \frac{1}{2\pi i} \int_C \sum_{p=0}^{n-1} ia^w w^{-k-1} dw $$

**3.3. Derivation of the second contour integral.** Using a generalization of Cauchy's integral formula (1.2) we first replace $y \to \log(a) + i2^{p+1}(y+1)$ then multiply both sides by $-2i(-1)^y e^{im2^{p+1}(y+1)}$ and take the infinite sum over $y \in [0, \infty)$ and $p \in [0, n-1]$ and simplify in terms of the Hurwitz-Lerch Zeta function to get

$$ (3.3) \qquad - \sum_{p=0}^{n-1} \frac{i 2^{k+1} (i 2^p)^k e^{im 2^{p+1}} \Phi(-e^{i 2^{p+1}m}, -k, 1 - i 2^{-p-1} \log(a))}{\Gamma(k+1)} \\ = -\frac{1}{2\pi i} \int_C \sum_{y=0}^{\infty} \sum_{p=0}^{n-1} 2ia^w w^{-k-1} e^{i(2^{p+1}(y+1)(m+w)+\pi y)} dw \\ = -\frac{1}{2\pi i} \int_C \sum_{p=0}^{n-1} \sum_{y=0}^{\infty} 2ia^w w^{-k-1} e^{i(2^{p+1}(y+1)(m+w)+\pi y)} dw \\ = \frac{1}{2\pi i} \int_C \sum_{p=0}^{n-1} (a^w w^{-k-1} \tan(2^p(m+w)) - ia^w w^{-k-1}) dw $$

from equation (1.232.1) in [6] where $Im(m + w) > 0$, $Re(m + w) > 0$, $n$ is a prime number in order for the sum to converge. We apply Tonelli's theorem for multiple sum, see page 189 in [7] as the summand is of bounded measure over the space $\mathbb{C} \times [0, n-1] \times [0, \infty)$.

**3.4. The Additional Contour Integral.** Using a generalization of Cauchy's integral formula (1.2) we first replace $y \to \log(a)$ and multiply both sides by $i$
simplify to get

$$ (3.4) \qquad -\sum_{p=0}^{n-1} \frac{i \log^k(a)}{\Gamma(k+1)} = -\frac{1}{2\pi i} \int_C \sum_{p=0}^{n-1} ia^w w^{-k-1} dw $$

## 4. Infinite Sum of the Contour Integral

**4.1. Derivation of the first contour integral.** We use the method in [4]. Using a generalization of Cauchy's integral formula (1.2) we first replace $y \to \log(a) + 2i(y+1)$ then multiply both sides by $-4ie^{2im(y+1)}$ and take the infinite sum over $y \in [0, \infty)$ and simplify in terms of the Hurwitz-Lerch Zeta function to get

$$ (4.1) \qquad \frac{i^{k-1} 2^{k+2} e^{2im} \Phi(e^{2im}, -k, 1 - \frac{1}{2}i \log(a))}{\Gamma(k+1)} \\ = -\frac{1}{2\pi i} \sum_{y=0}^{\infty} \int_C 4ia^w w^{-k-1} e^{2i(y+1)(m+w)} dw \\ = -\frac{1}{2\pi i} \int_C \sum_{y=0}^{\infty} 4ia^w w^{-k-1} e^{2i(y+1)(m+w)} dw \\ = \frac{1}{2\pi i} \int_C 2a^w w^{-k-1} \cot(m+w) + 2ia^w w^{-k-1} dw $$

from equation (1.232.1) in [6] where $Im(n(m+w)) > 0$ in order for the sum to converge.

**4.2. The Additional Contour Integral.** Using a generalization of Cauchy's integral formula (1.2) we first replace $y \to \log(a)$ and multiply both sides by $2i$ simplify to get

$$ (4.2) \qquad \frac{2i \log^k(a)}{\Gamma(k+1)} = \frac{1}{2\pi i} \int_C 2ia^w w^{-k-1} dw $$

**4.3. Derivation of the second contour integral.** We use the method in [4]. Using a generalization of Cauchy's integral formula (1.2) we first replace $y \to \log(a)+i2^{n+1}(y+1)$ then multiply both sides by $4ie^{im2^{n+1}(y+1)}$ and take the infinite sums over $y \in [0, \infty)$ and simplify in terms of the Hurwitz-Lerch zeta function to get

$$ (4.3) \qquad \frac{i2^{k+2} (i2^n)^k e^{im2^{n+1}} \Phi(e^{i2^{n+1}m}, -k, 1 - i2^{-n-1} \log(a))}{\Gamma(k+1)} \\ = \frac{1}{2\pi i} \sum_{y=0}^{\infty} \int_C 4ia^w w^{-k-1} e^{i2^{n+1}(y+1)(m+w)} dw $$
$$= \frac{1}{2\pi i} \int_C \sum_{y=0}^{\infty} 4ia^w w^{-k-1} e^{i2^{n+1}(y+1)(m+w)} dw$$

$$= \frac{1}{2\pi i} \int_C (-2a^w w^{-k-1} \cot(2^n(m+w)) - 2ia^w w^{-k-1}) dw$$

from equation (1.232.1) in [6] where $Im(n(m + w)) > 0$ in order for the sum to converge.

$$(4.4) \qquad -\frac{2i \log^k(a)}{\Gamma(k+1)} = -\frac{1}{2\pi i} \int_C 2ia^w w^{-k-1} dw$$

## 5. Finite Sum of the Hurwitz-Lerch Zeta function in terms of two Hurwitz-Lerch Zeta functions

THEOREM 5.1. For all $k, a, m \in \mathbb{C}$ and any prime number then,

$$(5.1) \quad -\sum_{p=0}^{n-1} i2^{k+1} (i2^p)^k e^{im2^{p+1}} \left( \Phi(-e^{i2^{p+1}m}, -k, 1 - i2^{-p-1}\log(a)) \right. \\ \qquad \left. + \Phi(e^{i2^{p+1}m}, -k, 1 - i2^{-p-1}\log(a)) \right)$$

$$= i2^{k+2} \left( (i2^n)^k e^{im2^{n+1}} \Phi(e^{i2^{n+1}m}, -k, 1 - i2^{-n-1}\log(a)) \right. \\ \qquad \left. -i^k e^{2im} \Phi(e^{2im}, -k, 1 - \frac{1}{2}i\log(a)) \right)$$

PROOF. Observe that the addition of the right-hand sides of equations (3.1) and (3.2) and (3.3) and (3.4), is equal to the addition of the right-hand sides of equations (4.1) and (4.2) and (4.3) and (4.4) so we may equate the left-hand sides and simplify using Example (74) in [2] and the Gamma function to yield the stated result. $\square$

EXAMPLE 5.2. The degenerate case.

$$(5.2) \qquad \sum_{p=0}^{n-1} 2 \csc(m2^{p+1}) = 2(\cot(m) - \cot(m2^n))$$

PROOF. Use equation (5.1) and set $k = 0$ and simplify using entry (4) below (64:12:7) in [8]. $\square$

## 6. Finite Trigonometric Products, Sums and Recurrence Identities

In this section we will evaluate equation (5.1) for various ranges of the parameters involved and apply the method in [9] to derive a recursive identity and finite products involving trigonometric functions.

EXAMPLE 6.1.

$$(6.1) \qquad \prod_{p=0}^{n-1} (\tan^2(2^{p-2}x) \tan(2^p x) \cot^3(2^{p-1}x))^{2^{p-2}}$$
$$ = \frac{1}{\sin^{\frac{3}{2}}\left(\frac{x}{2}\right)} \sin\left(\frac{x}{4}\right) \sqrt{\sin(x)} \sin^{2^{-n-1}}(2^{n-1}x) \sin^{-2^{-n-1}}(2^{n}x) \cos^{-2^{-n}}(2^{n-2}x) \\ \cos^{2^{-n}}(2^{n-1}x) \tan^{-2^{-n}}(2^{n-2}x) \tan^{2^{-n}}(2^{n-1}x) $$

PROOF. Use equation (5.1) and set $k = 1, a = 1, m = x$ and apply the method in section (8) in [9]. □

EXAMPLE 6.2. Recurrence identity with consecutive neighbours

$$ (6.2) \qquad \Phi(z, s, a) = 2^{1-s} z \Phi\left(z^2, s, \frac{a+1}{2}\right) + \Phi(-z, s, a) $$

PROOF. Use equation (5.1) and set $n = 1, a = e^{2i(a-1)}$, $k = -s, m = \log(z)/(2i)$ and simplify. □

EXAMPLE 6.3. Recurrence identity with consecutive neighbours

$$ (6.3) \qquad \Phi(-z, s, a) - \Phi(z, s, a) $$

$$ = 4^{-s} \left( -2^s z \left( \Phi\left(-z^2, s, \frac{a+1}{2}\right) + \Phi\left(z^2, s, \frac{a+1}{2}\right) \right) - 2z^3 \Phi\left(z^4, s, \frac{a+3}{4}\right) \right) $$

PROOF. Use equation (5.1) and set $n = 2, a = e^{2i(a-1)}$, $k = -s, m = \log(z)/(2i)$ and simplify. □

EXAMPLE 6.4. Difference of the first partial derivative of the Hurwitz-Lerch zeta function in terms of the log-gamma function.

$$ (6.4) \qquad \Phi'(-i, 0, a) - \Phi'(i, 0, a) = \frac{1}{2} i \log \left( \frac{16\Gamma\left(\frac{a+3}{4}\right)^4}{\Gamma\left(\frac{a+1}{4}\right)^4} \right) $$

PROOF. Use equation (6.2) and set $z = i$ and simplify in terms of the Hurwitz zeta function using entry (4) in Table below (64:12:7) in [8]. Next take the first partial derivative with respect to $s$ and set $s = 0$ and simplify in terms of the log-gamma function using equation (25.11.18) in [3]. □

EXAMPLE 6.5. Finite product involving quotient tangent and exponential cosine and sine functions.

$$ (6.5) \quad \prod_{p=0}^{n-1} e^{i \csc(2^{p+1}x) - i \csc(2^p x)} (\tan(2^{p-1}x) \cot(2^p x))^{\pi 2^{-p-1}} \\ = \sin^\pi\left(\frac{x}{2}\right) \sin^{-\pi}(x) \cos^{\pi(-2^{-n})}(2^{n-1}x) \cos^{\pi 2^{-n}}(2^nx) \tan^{\pi(-2^{-n})}(2^{n-1}x) \\ \qquad \qquad \qquad \qquad \qquad \tan^{\pi 2^{-n}}(2^nx) e^{-i(csc(x)-csc(2^nx))} $$

PROOF. Use equation (5.1) and set $k = 1, a = -1, m = x$ and apply the method in section (8) in [9]. Note $n$ is any prime number. □

EXAMPLE 6.6. Finite product involving quotient tangent and exponential cosine and sine functions.
$$
(6.6) \quad \prod_{p=0}^{n-1} e^{-2i(csc(2^p x)-csc(2^{p+1} x))} (\tan(2^{p-1}x) \cot(2^p x))^{\pi 2^{-p-1}} \\
\qquad = \sin^\pi\left(\frac{x}{2}\right) \sin^{-\pi}(x) \cos^\pi(-2^{-n}) (2^{n-1}x) \cos^{\pi 2^{-n}}(2^nx) \tan^\pi(-2^{-n}) (2^{n-1}x) \\
\qquad \qquad \qquad \qquad \tan^{\pi 2^{-n}}(2^nx) e^{-2i(csc(x)-csc(2^nx))}
$$

PROOF. Use equation (5.1) and set $k=1, a=i, m=x$ and apply the method in section (8) in [9]. Note $n$ is any prime number. $\square$

EXAMPLE 6.7. Finite product involving quotient tangent and exponential co-
sine and sine functions.

$$
(6.7) \quad \prod_{p=0}^{n-1} \left( \tan(2^p x) \cot(2^{p-1}x) \right)^{2^{-p-1}} e^{\frac{1}{2} \csc(2^p x) \sec(2^p x) - \csc(2^p x)} \\
\qquad = 2 \cos\left(\frac{x}{2}\right) \cos^{2^{-n}}(2^{n-1}x) \cos^{-2^{-n}}(2^nx) \tan^{2^{-n}}(2^{n-1}x) \tan^{-2^{-n}}(2^nx) \\
\qquad \qquad \qquad e^{\csc(2^nx) - \cot\left(\frac{x}{2}\right) + \cot(x)}
$$

PROOF. Use equation (5.1) and set $k=1, a=e, m=x$ and apply the method in section (8) in [9]. Note $n$ is any prime number. $\square$

EXAMPLE 6.8. Finite product involving the sine and cosine functions raised to powers.

$$
(6.8) \quad \prod_{p=0}^{n-1} \sin^{2^3-p} (2^{p-3}x) \sin^{-72^{1-p}} (2^{p-2}x) \sin^7 2^{-p} (2^{p-1}x) \sin^{-2-p} (2^px) \\
\qquad \qquad \cos^{-2^3-p} (2^{p-3}x) \cos^7 2^{1-p} (2^{p-2}x) \cos^{-72^{-p}} (2^{p-1}x) \cos^{2^{-p}} (2^px) \\
\qquad = \frac{1}{64} \cos^{12} \left(\frac{x}{4}\right) \sec^{16} \left(\frac{x}{8}\right) \sec^2 \left(\frac{x}{2}\right) \sin^{-2^{3-n}} (2^{n-3}x) \\
\qquad \qquad \sin^3 2^{2-n} (2^{n-2}x) \sin^{-2^{2-n}} (2^{n-1}x) \cos^{2^{3-n}} (2^{n-3}x) \cos^{-2^{4-n}} (2^{n-2}x) \\
\qquad \qquad \cos^5 2^{1-n} (2^{n-1}x) \cos^{-2^{1-n}} (2^nx) \tan^{-2^{2-n}} (2^{n-2}x) \tan^3 2^{1-n} (2^{n-1}x) \\
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \tan^{-2^{1-n}} (2^nx)
$$

PROOF. Use equation (5.1) and set $k=2, a=1, m=x$ and apply the method in section (8) in [9]. $\square$

EXAMPLE 6.9. A generalized case for the finite product of trigonometric func-
tions. For all $x, a, \beta \in \mathbb{C}, n \in \mathbb{Z}_+$,

$$
(6.9) \quad \prod_{p=0}^{n-1} e^{\csc(2^{p+1}x) - \csc\left(\frac{2^{p+1}x}{\beta}\right)} \left( \tan(2^p x) \cot\left(\frac{2^p x}{\beta}\right) \right)^{ia2^{-p-1}}
$$
$$ = e^{\cot(\frac{2^n x}{\beta}) - \cot(2^n x) - \cot(\frac{x}{\beta}) + \cot(x)} \left(\sin(x) \csc\left(\frac{x}{\beta}\right)\right)^{ia} \left(\csc(2^n x) \sin\left(\frac{2^n x}{\beta}\right)\right)^{ia2^{-n}} $$

PROOF. Use equation (5.1) and set $k = 2, a = 1, m = x$ and apply the method in section (8) in [9]. □

EXAMPLE 6.10. Finite product involving the ratio of tangent functions.

$$ (6.10) \quad \prod_{p=0}^{n-1} \left( \frac{\tan(2^p r)}{\tan(2^p m)} \right)^{2^{-p}} = e^{-2i(m-r)} \left( \frac{e^{i2^n(m-r)} \sin(2^n m)}{\sin(2^n r)} \right)^{2^{1-n}} \left( \frac{\sin(r)}{\sin(m)} \right)^2 $$

PROOF. Use equation (5.1) and form a second equation by replacing $m$ by $r$ and take their difference and set $k = -1, a = 1$ and simplify using entry (3) in Table below (64:12:7) in [8]. □

EXAMPLE 6.11. Extended Nielsen-Gauss Formula.

$$ (6.11) \quad \prod_{p=0}^{n-1} \Gamma\left(\frac{1}{2}(1+2^{-p}z)\right) = \frac{2^{2(-1+2^{-n})z} \pi^{n/2} \Gamma(1+z)}{\Gamma(1+2^{-n}z)} $$

PROOF. Use equation (5.1) and set $m = 0$ and simplify in terms of the Hurwitz zeta function using entry (4) in Table below (64:12:7) in [8]. Next take the first partial derivative with respect to $k$ and set $k = 0$ and simplify in terms of the log-gamma function using equation (25.11.18) in [3]. Next take the exponential function of both sides and simplify with replacing $a \to e^{zi}$. Similar forms are published in section (5.5) in [3] in the works by Sándor et. al [10] and equation (1.6.7) in [11], the infinite form by Knar, see equation (1.6) in [5] and equation (7) in [12]. □

The Nielsen form is given by by simply algebraic manipulation of equation (6.11) given by;

$$ (6.12) \quad \Gamma(1+z) = \frac{2^{2(2^{-n}-1)z}}{\pi^{n/2}} \Gamma(1+2^{-n}z) \prod_{p=0}^{n-1} \Gamma\left(\frac{1}{2}(1+2^{-p}z)\right) $$

## 7. Conclusion

In this paper, we have presented a method for deriving new finite trigonometric product formulae along with an interesting recurrence identity similar to those published in current literature with many more possible, using contour integration. The results presented were numerically verified for both real and imaginary and complex values of the parameters using Mathematica by Wolfram.

## References

[1] E. R. Hansen (1975) *A Table of Series and Products*. Prentice-Hall, Englewood Cliffs, N.J

[2] Edward Albert Bowser (1892) *A Treatise on Plane and Spherical Trigonometry And Its Applications to Astronomy and Geodesy*, Nature 40, 342 (1889). https://doi.org/10.1038/040342b0

[3] Olver, F.W.J.; Lozier, D.W.; Boisvert, R.F.; Clark, C.W. (Eds.) NIST Digital Library of Mathematical Functions; U.S. Department of Commerce, National Institute of Standards and Technology: Washington, DC, USA; Cambridge University Press: Cambridge, UK, 2010; With 1 CD-ROM (Windows, Macintosh and UNIX). MR 2723248 (2012a:33001).
[4] Reynolds, R.; Stauffer, A. A Method for Evaluating Definite Integrals in Terms of Special Functions with Examples. *Int. Math. Forum* **2020**, *15*, 235–244, doi:10.12988/imf.2020.91272

[5] Erdélyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F.G. *Higher Transcendental Functions*; McGraw-Hill Book Company, Inc.: New York, NY, USA; Toronto, ON, Canada; London, UK, 1953; Volume I.

[6] Gradshteyn, I.S.; Ryzhik, I.M. *Tables of Integrals, Series and Products*, 6th ed.; Academic Press: Cambridge, MA, USA, 2000.

[7] Gelca, Răzvan., Andreescu, Titu. *Putnam and Beyond*. Germany: Springer International Publishing, 2017.

[8] Oldham, K.B.; Myland, J.C.; Spanier, J. *An Atlas of Functions: With Equator, the Atlas Function Calculator*, 2nd ed.; Springer: New York, NY, USA, 2009.

[9] Reynolds, R., & Stauffer, A. (2022). *A Note on the Infinite Sum of the Lerch function*. European Journal of Pure and Applied Mathematics, 15(1), 158–168. https://doi.org/10.29020/nybg.ejpam.v15i1.4137

[10] J. Sándor and L. Tóth (1989) *A remark on the gamma function*. Elem. Math. 44 (3), pp. 73–76.

[11] N. Nielsen (1906a) *Handbuch der Theorie der Gammafunktion*. B. G. Teubner, Leipzig (German).

[12] Pearson, John. "Viète's Formula, Knar's Formula, and the Geometry of the Gamma Function.", The American Mathematical Monthly 125, no. 8 (2018): 704–14. https://www.jstor.org/stable/48661880.

(Robert Reynolds) DEPARTMENT OF MATHEMATICS AND STATISTICS, YORK UNIVERSITY, TORONTO, ON, CANADA, M3J1P3

Email address, Corresponding author: milver73@gmail.com