# A short note on a extended finite secant series

Robert Reynolds

► To cite this version:

Robert Reynolds. A short note on a extended finite secant series. 2023. hal-04140601

HAL Id: hal-04140601

https://hal.science/hal-04140601

Preprint submitted on 26 Jun 2023

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
1

# A short note on an extended finite secant series.

2

Robert Reynolds

**ABSTRACT.** In this paper, a summation formula for a general family of a finite secant sum has been extended by making use of a particularly convenient integration contour method. The main theorem derived from this approach is the finite sum involving the Hurwitz-Lerch zeta function. This theorem for particular values is used to derive the finite product of the fifth roots of the quotient product of the gamma function along with finite sums and functional equations involving trigonometric functions.

3

## 1. Introduction

The finite secant series and the Hurwitz-Lerch zeta function are mathematical concepts that are used in different areas of mathematics. The finite secant series is a generalization of the regular secant series, which is defined as the power series expansion of the secant function. The secant function is the reciprocal of the cosine function, and its series expansion involves powers of the variable raised to even positive integers. This series expands the secant function using powers of the variable raised to non-negative real numbers, which allows for a broader range of values. This series is typically used in mathematical analysis and approximation theory to approximate the secant function for a wider class of arguments.

The Hurwitz-Lerch zeta function, named after the Swiss mathematician Mathias Lerch, is a complex-valued function that generalizes several other mathematical functions, such as the Hurwitz zeta function and the polylogarithm function. This function has various applications in number theory, mathematical physics, and special functions theory. It appears in the study of Riemann zeta function, modular forms, and quantum statistical mechanics, among other areas. It has connections to other important functions, such as the polylogarithm function, and the Hurwitz zeta function for special cases of the parameters involved.

Finite secant series and contour integration have various applications in mathematics and physics. One area where they prove useful is in the calculation of special functions. By utilizing finite secant series and contour integrals, values of special functions like the gamma function, Bessel functions, and hypergeometric functions can be determined numerically. These functions can be expressed in terms of contour integrals, and the residue theorem is employed to evaluate them accurately.

22

23

24

25

26

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

<footnote>Key words and phrases. Hurwitz-Lerch zeta function, secant function, gamma function.</footnote>
27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

Another significant application lies in the evaluation of real integrals that are challenging to compute using standard methods. By combining contour integration techniques with finite secant series, complex integrals can be simplified. Appropriate contour choices and utilization of the integrand's properties facilitate the calculation process, ultimately leading to finite secant series representations.

Finite secant series also offer an alternative representation for certain series, aiding in their analysis and summation. By expressing a given series as a finite secant series, closed-form expressions can potentially be derived, patterns can be identified, and convergence properties can be explored.

Contour integration techniques, in conjunction with finite secant series, can be applied to solve differential equations. Transforming a given differential equation into an integral equation through contour integration allows for the utilization of finite secant series methods in solving it. This approach is particularly advantageous when dealing with linear differential equations that have special functions as solutions.

In the realm of theoretical physics, contour integration plays a vital role in the formulation and evaluation of scattering amplitudes in quantum field theory. Contour integration techniques enable the calculation of loop integrals encountered in perturbative expansions. Finite secant series offer compact and efficient representations of these loop integrals, assisting in higher-order calculations in quantum field theory. While there are numerous references on the applications of finite secant series and contour integration, here are a few notable sources that provide detailed information are recorded in [1, 2, 3, 4] In this work, we apply the contour integral method from [5], to the finite sum of the secant function where the general derivation is given on page 90 in [6], resulting in

$$ (1.1) \qquad -\frac{1}{2\pi i} \int_C \sum_{p=0}^{n} 2 \times 5^{-p} a^w w^{-k-1} \left( -5 \cos(2 \ 5^{p-1}(m+w)) + 5 \cos(4 \ 5^{p-1}(m+w)) + 2 \right) \\ \qquad \qquad \qquad \qquad \sec(5^p(m+w)) dw \\ \qquad = \frac{1}{2\pi i} \int_C 5^{-n} a^w w^{-k-1} \left( \sec(5^n(m+w)) - 5^{n+1} \sec\left(\frac{m+w}{5}\right) \right) dw $$

where $a, m, k \in \mathbb{C}$, $Re(m+w) > 0$, $n \in \mathbb{Z}^+$. Using equation (1.1) the main Theorem to be derived and evaluated is given by

$$ (1.2) \quad \sum_{p=0}^{n} 5^{-p} (i5^p)^k e^{im5^{p-1}} \left( \begin{aligned} & -5\Phi\left(-e^{2i5^p m}, -k, \frac{1}{10} - \frac{1}{2}i5^{-p} \log(a)\right) \\ & +5e^{2im5^{p-1}} \Phi\left(-e^{2i5^p m}, -k, \frac{3}{10} - \frac{1}{2}i5^{-p} \log(a)\right) \\ & -4e^{4im5^{p-1}} \Phi\left(-e^{2i5^p m}, -k, \frac{1}{2} - \frac{1}{2}i5^{-p} \log(a)\right) \\ & +5e^{6im5^{p-1}} \Phi\left(-e^{2i5^p m}, -k, \frac{7}{10} - \frac{1}{2}i5^{-p} \log(a)\right) \\ & -5e^{8im5^{p-1}} \Phi\left(-e^{2i5^p m}, -k, \frac{9}{10} - \frac{1}{2}i5^{-p} \log(a)\right) \end{aligned} \right) $$
$$= 5^{-k-n} \left( (i5^{n+1})^k e^{im5^n} \Phi \left( -e^{2i5^n m}, -k, \frac{1}{2} - \frac{1}{2}i5^{-n} \log(a) \right) -i^k e^{\frac{im}{5}} 5^{n+1} \Phi \left( -e^{\frac{2im}{5}}, -k, \frac{1}{2} - \frac{5}{2}i \log(a) \right) \right)$$

where the variables $k, a, m$ are general complex numbers and $n$ is any positive integer. This new expression is then used to derive special cases in terms of trigonometric functions. The derivations follow the method used by us in [5]. This method involves using a form of the generalized Cauchy's integral formula given by

$$(1.3) \qquad \frac{y^k}{\Gamma(k+1)} = \frac{1}{2\pi i} \int_C \frac{e^{wy}}{w^{k+1}} dw,$$

where $y, w \in \mathbb{C}$ and $C$ is in general an open contour in the complex plane where the bilinear concomitant [5] is equal to zero at the end points of the contour. This method involves using a form of equation (1.3) 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 (1.3) 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 [7] where $\Phi(z, s, v)$ is the Hurwitz-Lerch zeta function which is a generalization of the Hurwitz zeta $\zeta(s, v)$ and Polylogarithm function $\mathrm{Li}_n(z)$. 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 $|z|< 1, v \neq 0, -1, -2, -3, ..$, and is continued analytically by its integral representation given by

$$(2.2) \qquad \Phi(z, s, v) = \frac{1}{\Gamma(s)} \int_0^\infty \frac{t^{s-1} e^{-vt}}{1 - ze^{-t}} dt = \frac{1}{\Gamma(s)} \int_0^\infty \frac{t^{s-1} e^{-(v-1)t}}{e^t - z} dt$$

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. Contour Integral Representation for the Finite Sum of the Hurwitz-Lerch zeta Functions

In this section we derive the contour integral representations of the left-hand side and right-hand side of equation (1.1) in terms of the Hurwitz-Lerch zeta and trigonometric functions.

**3.1. Derivation of the generalized Hurwitz-Lerch contour integral for the secant function.** We use the method in [5]. Using equation (1.3) we first replace $\log(a) + ib(2y+1)$ and multiply both sides by $-2i(-1)^y e^{ibm(2y+1)}$ then take the finite and infinite sums over $p \in [0, n]$ and $y \in [0, \infty)$ and simplify in terms of the Hurwitz-Lerch zeta function to get

$$(3.1) \qquad -\sum_{p=0}^{n} \frac{i2^{k+1}(ib)^k e^{ibm}\Phi\left(-e^{2ibm}, -k, \frac{b-i\log(a)}{2b}\right)}{\Gamma(k+1)}$$
$$
\begin{align*}
&= \frac{1}{2\pi i} \sum_{y=0}^{\infty} \sum_{p=0}^{n} \int_C (-1)^y a^w w^{-k-1} e^{ib(2y+1)(m+w)} dw \\
&= \frac{1}{2\pi i} \int_C \sum_{p=0}^{n} \sum_{y=0}^{\infty} (-1)^y a^w w^{-k-1} e^{ib(2y+1)(m+w)} dw \\
&= -\frac{1}{2\pi i} \int_C \sum_{p=0}^{n} ia^w w^{-k-1} \sec(b(m+w)) dw
\end{align*}
$$

59 from equation (1.232.2) in [8] where Re(w + m) > 0 and Im(m + w) > 0 in order
60 for the sums to converge. We apply Tonelli's theorem for multiple sums, see page
61 177 in [9] as the summands are of bounded measure over the space ℂ × [0, n] × [0, ∞).

3.2. Derivation of a generalized Hurwitz-Lerch zeta function in terms of the product of the cosine and secant function contour integral. We use the method in [5]. Using a generalization of Cauchy's integral formula (1.3) we first replace y by log(a) + ix + y then multiply both sides by e^{mxi} then form a second equation by replacing x by -x and add both equations to get

$$
(3.2) \quad \frac{e^{-imx} (e^{2imx} (\log(a) + ix + y)^k + (\log(a) - ix + y)^k)}{\Gamma(k+1)} \\
\qquad = \frac{1}{2\pi i} \int_C 2w^{-k-1} e^{w(\log(a)+y)} \cos(x(m+w)) dw
$$

Next we replace $y$ by $ib(2y + 1)$ and multiply both sides by $(-1)^y e^{ibm(2y+1)}$ and
take the infinite sum over $y \in [0, \infty)$ and simplify in terms of the Hurwitz-Lerch
zeta function to get

$$
\begin{align*}
& (3.3) \frac{2^k (ib)^k e^{im(b-x)} \left( \Phi\left(-e^{2ibm}, -k, \frac{b-x-i \log(a)}{2b}\right) + e^{2imx} \Phi\left(-e^{2ibm}, -k, \frac{b+x-i \log(a)}{2b}\right) \right)}{\Gamma(k+1)} \\
&= \frac{1}{2\pi i} \sum_{y=0}^{\infty} \int_C 2(-1)^y a^w w^{-k-1} e^{ib(2y+1)(m+w)} \cos(x(m+w)) dw \\
&= \frac{1}{2\pi i} \int_C \sum_{y=0}^{\infty} 2(-1)^y a^w w^{-k-1} e^{ib(2y+1)(m+w)} \cos(x(m+w)) dw \\
&= \frac{1}{2\pi i} \int_C a^w w^{-k-1} \sec(b(m+w)) \cos(x(m+w)) dw
\end{align*}
$$

62 from equation (1.232.2) and (1.411.3) in [8] where Re(w+m) > 0 and Im(m + w) >
63 0 in order for the sums to converge. We apply Tonelli's theorem for sums and
64 integrals, see page 177 in [9] as the summand and integral are of bounded measure
65 over the space $\mathbb{C} \times [0, \infty)$.

3.3. Derivation of the contour integrals. In this section we will use equa-
tions (3.1) and (3.3) by simple substitution to derive the contour integrals in equa-
tion (1.1).
3.3.1. *Derivation of the left-hand first contour integral.* Use equation (3.1) and replace $b$ by $5^p$ then multiply both sides by $-4i5^{-p}$ and take the finite sum over $p \in [0, n]$ to get;

$$ (3.4) \quad - \sum_{p=0}^{n} \frac{2^{k+3} 5^{-p} (i5^p)^k e^{im5^p} \Phi(-e^{2i5^p m}, -k, \frac{1}{2} 5^{-p} (5^p - i \log(a)))}{\Gamma(k+1)} \\ = -\frac{1}{2\pi i} \int_C \sum_{p=0}^{n} 4 \times 5^{-p} a^w w^{-k-1} \sec(5^p(m+w)) dw $$

3.3.2. *Derivation of the left-hand second contour integral.* Use equation (3.3) and replace $x$ by $2 \ 5^{p-1}$, $b$ by $5^p$ then multiply both sides by $2 \ 5^{1-p}$ and take the finite sum over $p \in [0, n]$ to get;

$$ (3.5) \quad \sum_{p=0}^{n} \frac{1}{\Gamma(k+1)} 2^{k+1} 5^{1-p} (i5^p)^k e^{im(5^p-2 \ 5^{p-1})} \left( \Phi\left(-e^{2i5^p m}, -k, \frac{1}{2} 5^{-p} \left(-i \log(a) - 2 \ 5^{p-1} + 5^p\right)\right) \right. \\ \left. \qquad + e^{4im5^{p-1}} \Phi\left(-e^{2i5^p m}, -k, \frac{1}{2} 5^{-p} \left(-i \log(a) + 2 \ 5^{p-1} + 5^p\right)\right) \right) \\ = \frac{1}{2\pi i} \int_C \sum_{p=0}^{n} 2 \times 5^{1-p} a^w w^{-k-1} \cos(2 \ 5^{p-1}(m+w)) \sec(5^p(m+w)) dw $$

3.3.3. *Derivation of the left-hand third contour integral.* Use equation (3.3) and replace $x$ by $4 \times 5^{p-1}$, $b$ by $5^p$ then multiply both sides by $-2 \times 5^{1-p}$ and take the finite sum over $p \in [0, n]$ to get;

$$ (3.6) \quad - \sum_{p=0}^{n} \frac{1}{\Gamma(k+1)} 2^{k+1} 5^{1-p} (i5^p)^k e^{im(5^p-4 \ 5^{p-1})} \left( \Phi\left(-e^{2i5^p m}, -k, \frac{1}{2} 5^{-p} \left(-i \log(a) - 4 \ 5^{p-1} + 5^p\right)\right) \right. \\ \left. \qquad + e^{8im5^{p-1}} \Phi\left(-e^{2i5^p m}, -k, \frac{1}{2} 5^{-p} \left(-i \log(a) + 4 \ 5^{p-1} + 5^p\right)\right) \right) \\ = -\frac{1}{2\pi i} \int_C \sum_{p=0}^{n} 25^{1-p} a^w w^{-k-1} \cos(4 \ 5^{p-1}(m+w)) \sec(5^p(m+w)) dw $$

3.3.4. *Derivation of the right-hand first contour integral.* Use equation (3.1) and replace $b$ by $1/5$ then multiply both sides by $5/i$ to get;

$$ (3.7) \quad - \frac{i^k 2^{k+1} 5^{1-k} e^{\frac{im}{5}} \Phi\left(-e^{\frac{2im}{5}}, -k, \frac{5}{2}\left(\frac{1}{5} - i \log(a)\right)\right)}{\Gamma(k+1)} \\ = -\frac{1}{2\pi i} \int_C 5a^w w^{-k-1} \sec\left(\frac{m+w}{5}\right) dw $$

3.3.5. *Derivation of the right-hand second contour integral.* Use equation (3.1) and replace $b$ by $5^n$ then multiply both sides by $-5^n/i$ to get;

$$ (3.8) \quad \frac{2^{k+1} 5^{-n} (i5^n)^k e^{im5^n} \Phi(-e^{2i5^n m}, -k, \frac{1}{2} 5^{-n} (5^n - i \log(a)))}{\Gamma(k+1)} $$
69

$$ = \frac{1}{2\pi i} \int_C 5^{-n} a^w w^{-k-1} \sec(5^n(m+w)) dw $$

## 4. Evaluations and derivations involving special functions

70

71

72

73

74

The Lerch transcendent or Hurwitz-Lerch zeta function is a special function that appears in number theory, complex analysis, and other areas of mathematics. Its symbol is $\Phi(z, s, a)$. The Hurwitz-Lerch zeta function is derived in this section by determining the function's derivatives with respect to the given variables. We also derive gamma function-related infinite and finite products.

**THEOREM 4.1.** For all $k, a, m \in \mathbb{C}$ then,

$$ \begin{aligned} & (4.1) \quad \sum_{p=0}^{n} 5^{-p} (i5^p)^k e^{im5^{p-1}} \left( \begin{aligned}[t] & -5\Phi\left(-e^{2i5^p m}, -k, \frac{1}{10} - \frac{1}{2}i5^{-p} \log(a)\right) \\ & +5e^{2im5^{p-1}} \Phi\left(-e^{2i5^p m}, -k, \frac{3}{10} - \frac{1}{2}i5^{-p} \log(a)\right) \\ & -4e^{4im5^{p-1}} \Phi\left(-e^{2i5^p m}, -k, \frac{1}{2} - \frac{1}{2}i5^{-p} \log(a)\right) \\ & +5e^{6im5^{p-1}} \Phi\left(-e^{2i5^p m}, -k, \frac{7}{10} - \frac{1}{2}i5^{-p} \log(a)\right) \\ & -5e^{8im5^{p-1}} \Phi\left(-e^{2i5^p m}, -k, \frac{9}{10} - \frac{1}{2}i5^{-p} \log(a)\right) \end{aligned} \right) \\ & = 5^{-k-n} \left( (i5^{n+1})^k e^{im5^n} \Phi\left(-e^{2i5^n m}, -k, \frac{1}{2} - \frac{1}{2}i5^{-n} \log(a)\right) \right. \\ & \qquad \left. -i^k e^{\frac{im}{5}} 5^{n+1} \Phi\left(-e^{\frac{2im}{5}}, -k, \frac{1}{2} - \frac{5}{2}i \log(a)\right) \right) \end{aligned} $$

75

76

77

**PROOF.** Since the addition of the right-hand side of equations (3.4), (3.5) and (3.6) is equal to the addition of right-hand sides of equations (3.8) and (3.7) relative to equation (1.1), we may equate the left-hand sides to yield the stated result. $\square$

**EXAMPLE 4.2.** The degenerate case.

$$ \begin{aligned} & (4.2) \quad \sum_{p=0}^{n} 5^{-p} \left( 5 \cos (2m5^{p-1}) - 5 \cos (4m5^{p-1}) - 2 \right) \sec (m5^p) \\ & \qquad = \frac{1}{2} \left( 5^{-n} \sec (m5^n) - 5 \sec \left( \frac{m}{5} \right) \right) \end{aligned} $$

78

79

**PROOF.** Use equation (4.1) and set $k = 0$ and simplify using entry (2) in Table below (64:12:7) in [10]. $\square$

**EXAMPLE 4.3.** Finite product of the fifth roots of product quotient gamma functions.

(4.3)

$$ \prod_{p=0}^{n} \sqrt[5p]{\frac{\Gamma\left(\frac{5-pa}{4} + \frac{1}{20}\right) \Gamma\left(\frac{5-pa}{4} + \frac{9}{20}\right) \Gamma\left(\frac{5-pa}{4} + \frac{13}{20}\right) \Gamma\left(\frac{5-pa}{4} + \frac{17}{20}\right) \left(\frac{\Gamma\left(\frac{1}{4}\left(5-pa+1\right)\right)}{\Gamma\left(\frac{1}{4}\left(5-pa+3\right)\right)}\right)^{4/5}}{\Gamma\left(\frac{5-pa}{4} + \frac{3}{20}\right) \Gamma\left(\frac{5-pa}{4} + \frac{7}{20}\right) \Gamma\left(\frac{5-pa}{4} + \frac{11}{20}\right) \Gamma\left(\frac{5-pa}{4} + \frac{19}{20}\right)}} $$
$$= \frac{5^{\frac{5}{8}-\frac{5-n}{8}} \Gamma(\frac{5a}{4}+\frac{1}{4}) \left( \frac{\Gamma(\frac{1}{4}(5-n)a+3)}{\Gamma(\frac{1}{4}(5-n)a+1)} \right)^{5-n-1}}{\Gamma(\frac{5a}{4}+\frac{3}{4})}$$

PROOF. Use equation (4.1) and set $m = 0$ and simplify to yield the Hurwitz zeta function using entry (4) in Table below (64:12:7) in [10]. Next we take the first partial derivative with respect to $k$ and set $k = 0$ and simplify in terms of the log-gamma function, using equation (64:10:2) in [10]. Next we take the exponential function of both sides and simplify in terms of the gamma function. The $\log \Gamma(z)$ is analytic throughout the complex $z$ plane, except for a single branch cut discontinuity along the negative real axis. The $\log(\Gamma(z))$ has a more complex branch cut structure. Similar forms of this interesting formula are recorded in [11, 12, 13, 14, 15]. □

4.1. The $\sqrt{5}$ and the Gamma function $\Gamma(z)$. The $\sqrt{5}$ finds its application in diverse fields, including mathematics, physics, engineering, finance, and computer science. In mathematics, it is extensively used in equations and formulas, particularly in geometry and trigonometry. It enables the calculation of distances, angles, and proportions in various geometric shapes and calculations. In physics, the $\sqrt{5}$ plays a role in determining quantities related to wave propagation, resonance, and harmonic oscillators. Equations that describe wave behavior, such as those concerning sound waves and electromagnetic waves, incorporate the $\sqrt{5}$. Engineers frequently employ the $\sqrt{5}$ in vibrations and structural mechanics calculations. It is significant in determining natural frequencies, resonance points, and damping ratios of mechanical systems. In the realm of finance, the $\sqrt{5}$ is employed in diverse calculations associated with risk and volatility. For instance, it is used in calculating the standard deviation, which measures the dispersion of data points from their mean. Additionally, financial models like the Black-Scholes formula for option pricing utilize the $\sqrt{5}$. Computer scientists also leverage the $\sqrt{5}$ in algorithms and computational techniques. It is commonly utilized in numerical analysis and optimization algorithms, where iterative methods may depend on square root calculations to converge towards desired solutions.

EXAMPLE 4.4. The product of quotient of gamma functions in terms of the square root of 5. The square root of 5 and its applications are recorded in the works of Campbell [16], Magnus et al. [17] and Lewis [18].

$$(4.4) \quad \frac{\Gamma(\frac{a}{4} + \frac{1}{20}) \Gamma(\frac{a}{4} + \frac{1}{4}) \Gamma(\frac{a}{4} + \frac{9}{20}) \Gamma(\frac{a}{4} + \frac{13}{20}) \Gamma(\frac{a}{4} + \frac{17}{20}) \Gamma(\frac{5a}{4} + \frac{3}{4})}{\Gamma(\frac{a}{4} + \frac{3}{20}) \Gamma(\frac{a}{4} + \frac{7}{20}) \Gamma(\frac{a}{4} + \frac{11}{20}) \Gamma(\frac{a}{4} + \frac{3}{4}) \Gamma(\frac{a}{4} + \frac{19}{20}) \Gamma(\frac{5a}{4} + \frac{1}{4})} \\ = \int_{-\infty}^{\infty} \frac{e^{-\frac{x^2}{5}}}{\sqrt{\pi}} dx = \sqrt{5}$$

PROOF. Use equation (4.3) and set $n = 0$ and simplify using equation (3.323.2) in [8]. □

EXAMPLE 4.5. The infinite product of the fifth roots of product quotient gamma functions. This infinite product of the gamma function has also be recorded in the works of Davis [19], Duke [20], Adamchik [21] and Borwein et al. [22]. The right-hand side of equation (4.5) can also be used in the solution of Legendre and related functions and their differential equation form see equation (14.2.11) in [23].
The right-hand side of equation (14.2.11) can be replaced by an infinite product by setting $u = \frac{5a}{4} + \frac{1}{2}$, $v = -\frac{1}{4}$ where the resulting product form is given by equation (4.5).

$$ (4.5) \qquad \prod_{p=0}^{\infty} \sqrt[5p]{\frac{\Gamma\left(\frac{5-pa}{4} + \frac{1}{20}\right) \Gamma\left(\frac{5-pa}{4} + \frac{9}{20}\right) \Gamma\left(\frac{5-pa}{4} + \frac{13}{20}\right) \Gamma\left(\frac{5-pa}{4} + \frac{17}{20}\right) \left(\frac{\Gamma\left(\frac{1}{4}\left(5-pa+1\right)\right)}{\Gamma\left(\frac{1}{4}\left(5-pa+3\right)\right)}\right)^{4/5}}{\Gamma\left(\frac{5-pa}{4} + \frac{3}{20}\right) \Gamma\left(\frac{5-pa}{4} + \frac{7}{20}\right) \Gamma\left(\frac{5-pa}{4} + \frac{11}{20}\right) \Gamma\left(\frac{5-pa}{4} + \frac{19}{20}\right)} \\ = \int_0^\infty \frac{5^{5/8} x^{\frac{5a}{4}-\frac{3}{4}} (x+1)^{-\frac{5a}{4}-\frac{3}{4}}}{\sqrt{\pi}} = \frac{5^{5/8} \Gamma\left(\frac{5a}{4} + \frac{1}{4}\right)}{\Gamma\left(\frac{1}{4}\left(5a+3\right)\right)} $$

PROOF. Use equation (4.3) and take the limit as $n \to \infty$ and simplify the right-hand side using equation (2) in [24] □

<figure><img src="image_2.png" /><figcaption>FIGURE 1. Plot of $\frac{5^{5/8}\Gamma(\frac{1}{4}(5a+1))}{\Gamma(\frac{1}{4}(5a+3))}$, $z \in \mathbb{R}$.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>FIGURE 2. Plot of $\mathrm{Re}\left(\frac{5^{5/8}\Gamma(\frac{1}{4}(5a+1))}{\Gamma(\frac{1}{4}(5a+3))}\right)$, $z \in \mathbb{C}$.</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>FIGURE 3. Plot of Im $\left(\frac{5^{5/8}\Gamma(\frac{1}{4}(5a+1))}{\Gamma(\frac{1}{4}(5a+3))}\right)$, $z \in \mathbb{C}$.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>FIGURE 4. Plot of Abs $\left(\frac{5^{5/8}\Gamma(\frac{1}{4}(5a+1))}{\Gamma(\frac{1}{4}(5a+3))}\right)$, $z \in \mathbb{C}$.</figcaption></figure>

EXAMPLE 4.6. A functional equation for the Hurwitz-Lerch zeta function. This type of equation is also recoded in the work by Oberhettinger [25].

$$ (4.6) \quad \Phi(z, s, a) \\ = 5^{-s} \left( \Phi\left(z^5, s, \frac{a}{5}\right) + z \left( \Phi\left(z^5, s, \frac{a+1}{5}\right) + z \left( \Phi\left(z^5, s, \frac{a+2}{5}\right) + z \left( \Phi\left(z^5, s, \frac{a+3}{5}\right) \right. \right. \\ \left. \left. + z\Phi\left(z^5, s, \frac{a+4}{5}\right) \right) \right) \right) $$
111 PROOF. Use equation (4.1) and set $n = 0$ and simplify. Next replace $m \to 5 \log(z)/(2i)$, $k \to -s$, $a \to e^{ai}$ and simplify. Next replace $a \to 2(a - 1/2)/5$, $z = -z$ and simplify. $\square$

113

EXAMPLE 4.7. A finite sum involving sine and cosine functions.

$$ (4.7) \quad \sum_{p=0}^{n} \left( \begin{aligned} & 9 \sin(5^{p-1}x) - 7 \sin(3 \ 5^{p-1}x) - 3 \sin(7 \ 5^{p-1}x) \\ & \qquad + \sin(9 \ 5^{p-1}x) + 4 \sin(5^p x) \sec^2(5^p x) \\ & = - \sec\left(\frac{x}{5}\right) \left(\sin\left(\frac{x}{5}\right) \sin(5^n x) + 1\right) \sec^2(5^n x) \left(\sec\left(\frac{x}{5}\right) \sin(5^n x) - \tan\left(\frac{x}{5}\right)\right) \end{aligned} \right) $$

114

PROOF. Use equation (4.1) and set $k = 1, a = 1, m = x$ and simplify. $\square$

EXAMPLE 4.8. Finite product involving the fifth roots of the product of quotient sine functions.

$$ (4.8) \quad \prod_{p=0}^{n} \left( \frac{\left( \frac{\sin\left(\frac{1}{2}5^{p-1}x\right) + 1}{\sin\left(\frac{1}{2}5^{p-1}x\right) - 1} \right) \left( \frac{\sin(5^{p-1}x) - 1}{\sin(5^{p-1}x) + 1} \right)}{\left( \frac{-2 \sin\left(\frac{1}{2}5^{p-1}x\right) + 2 \cos(5^{p-1}x) - 1}{2 \sin\left(\frac{1}{2}5^{p-1}x\right) + 2 \cos(5^{p-1}x) - 1} \right) \left( \frac{2 \sin(5^{p-1}x) + 2 \cos(2 \ 5^{p-1}x) - 1}{-2 \sin(5^{p-1}x) + 2 \cos(2 \ 5^{p-1}x) - 1} \right)} \right)^{\frac{12 \times 5^{-2p}}{5^{2p}}} \\ = \frac{\left( \frac{\left( \frac{\sin\left(\frac{5^n}{2}\right) - 1}{\sin\left(\frac{5^n}{2}\right) + 1} \right) \left( \sin(5^n x) + 1 \right)}{\left( \frac{\sin\left(\frac{5^n}{2}\right) + 1}{\sin\left(\frac{5^n}{2}\right) - 1} \right) \left( \sin(5^n x) - 1 \right)} \right)^{\frac{5^{-2n}}{2}} \left( \cos\left(\frac{3x}{20}\right) - \sin\left(\frac{x}{20}\right) \right)^{25}}{\left( \sin\left(\frac{x}{20}\right) + \cos\left(\frac{3x}{20}\right) \right)^{25}} $$

115

116

PROOF. Use equation (4.1) and set $k = 1, a = 1, m = x$ and simplify using the method in section (8.1) in [26]. $\square$

EXAMPLE 4.9. The infinite case of the product involving the fifth roots of the product of quotient sine functions. Note here $-1 < Re(x) < 1, -1 < Im(x) < 1$.

$$ (4.9) \quad \prod_{p=0}^{\infty} \left( \frac{\left( \frac{\sin\left(\frac{1}{2}5^{p-1}x\right) + 1}{\sin\left(\frac{1}{2}5^{p-1}x\right) - 1} \right) \left( \frac{\sin(5^{p-1}x) - 1}{\sin(5^{p-1}x) + 1} \right)}{\left( \frac{-2 \sin\left(\frac{1}{2}5^{p-1}x\right) + 2 \cos(5^{p-1}x) - 1}{2 \sin\left(\frac{1}{2}5^{p-1}x\right) + 2 \cos(5^{p-1}x) - 1} \right) \left( \frac{2 \sin(5^{p-1}x) + 2 \cos(2 \ 5^{p-1}x) - 1}{-2 \sin(5^{p-1}x) + 2 \cos(2 \ 5^{p-1}x) - 1} \right)} \right)^{\frac{12 \times 5^{-2p}}{5^{2p}}} \\ = \frac{\left( \cos\left(\frac{3x}{20}\right) - \sin\left(\frac{x}{20}\right) \right)^{25}}{\left( \sin\left(\frac{x}{20}\right) + \cos\left(\frac{3x}{20}\right) \right)^{25}} $$

117

118

PROOF. Use equation (4.8) and applying the limit as $n \to \infty$ along with analyzing a plot of the right-hand side we can write down the stated result. $\square$
<figure><img src="image_6.png" /><figcaption>FIGURE 5. Plot of $\left( \frac{\left( \sin\left(\frac{5^n x}{2}\right) - 1 \right) \left( \sin(5^n x) + 1 \right)}{\left( \sin\left(\frac{5^n x}{2}\right) + 1 \right) \left( \sin(5^n x) - 1 \right)} \right)^{\frac{5-2n}{2}}$, $x \in \mathbb{R}$.</figcaption></figure>

5. Conclusion

In this research paper, we presented formulas using trigonometric and special functions, and methods that can obtain formulas similar to those previously published previously. Our method is based on contour integration. The results of numerical calculations for a variety of parameters in mathematical expressions, including real, imaginary, and complex numbers, were verified with the help of the Wolfram software Mathematica.

References

[1] Dettman, John W.. "Applied Complex Variables", Dover Books on Mathematics, Courier Corporation, New York, 481 pages, (1984).

[2] Kreyszig, Erwin. *Advanced Engineering Mathematics*. New York:Wiley, 1983.

[3] Courant, Richard and Hilbert, David. *Methods of Mathematical Physics. Vol. 1*. New York: Wiley, 1989.

[4] Ahlfors, Lars V. *Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable*. 3rd ed. New York, NY: McGraw-Hill, 1979. ISBN: 9780070006577.

[5] 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

[6] Hobson, Ernest William. *A Treatise on Plane Trigonometry*, United Kingdom: University Press, 1897.

[7] 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.

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

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

[10] 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.

[11] Joseph Knar, *Die harmonischen reihen*, Archiv der Mathematik und Physik 41–42 (1864) p. 360.

[12] Nielsen, Niels. "Handbuch der Theorie der Gammafunktion.", Leipzig: Teubner, 1906,

[13] G. E. Andrews, R. Askey, and R. Roy (1999) *Special Functions. Encyclopedia of Mathematics and its Applications*, Vol. 71, Cambridge University Press, Cambridge.
153

154

155

156

157

158

159

160

161

162

163

164

165

166

167

168

169

170

171

172

173

174

175

176

177

178

179

180

181

182

183

[14] Raimundas VIDUNAS, *Expressions for values of the Gamma function*, Kyushu Journal of Mathematics, 2005, Volume 59, Issue 2, Pages 267-283, Released on J-STAGE October 14, 2005, Online ISSN 1883-2032, Print ISSN 1340-6116, https://doi.org/10.2206/kyushujm.59.267

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

[16] Campbell, R. *Les intégrales eulériennes et leurs applications*. Paris: Dunod, 1966.

[17] Magnus, W. and Oberhettinger, F. *Formulas and Theorems for the Special Functions of Mathematical Physics*. New York: Chelsea, 1949.

[18] P A J Lewis (2008). *Essential Mathematics 9*. Ratna Sagar. p. 24. ISBN 9788183323673.

[19] Davis, Philip J. "Leonhard Euler's Integral: A Historical Profile of the Gamma Function: In Memoriam: Milton Abramowitz.", The American Mathematical Monthly 66, no. 10 (1959): 849-69. https://doi.org/10.2307/2309786.

[20] Duke, W.; Imamoglu, Ö. (2006). "Special values of multiple gamma functions", Journal de Théorie des Nombres de Bordeaux. 18 (1): 113-123. doi:10.5802/jtnb.536. MR 2245878.

[21] Adamchik, V.S. *The Multiple Gamma Function and Its Application to Computation of Series*. Ramanujan J 9, 271-288 (2005). https://doi.org/10.1007/s11139-005-1868-3

[22] J. M. BORWEIN, I. J. ZUCKER, *Fast evaluation of the gamma function for small rational fractions using complete elliptic integrals of the first kind*, IMA Journal of Numerical Analysis, Volume 12, Issue 4, October 1992, Pages 519-526, https://doi.org/10.1093/imanum/12.4.519

[23] 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).

[24] A. Erdélyi, F. G. Tricomi *The asymptotic expansion of a ratio of gamma functions*, Pacific Journal of Mathematics, Pacific J. Math. 1(1), 133-142, (1951)

[25] Oberhettinger, F. "Note on the Lerch zeta function.", Pacific J. Math. 6 (1) 117 - 120, 1956

[26] 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

184

185

186

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

*Email address, Corresponding author: milver73@gmail.com*