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The context of this question is an investigation into the relationship between different representations of the error terms in staircase and prime counting functions such as $S(x)=\lfloor x\rfloor$ and the second Chebyshev function $\psi(x)$.

The following plot illustrates the function $S(x)-\left(x-\frac{1}{2}\right)$ in blue, a Fourier series representation of the error term in green, and a hypergeometric series represention of the error term in red.

Representations of Error Term in Staircase Function $S(x)$

The following plot illustrates the function $\psi(x)-(x-\log(2\pi))$ in blue, the zeta zero series representation of error term in orange, a Fourier series representation of the error term in green, and a hypergeometric series represention of the error term in red. With respect to the zeta zero series representation, the sum is over the first 100 zeta zero contributions in von Mangoldt's explicit formula for $\psi(x)$. As x increases, the hypergeometric series representation degrades faster than the other representations because it's the most complex function and consequently it was necessary to restrict the upper evaluation limits in order to obtain a reasonable evaluation time.

Representations of Error Term in Second Chebyshev Function $\psi(x)$

The hypergeometric series representation consists of terms with characteristics such as the following. The terms are actually more complex, but I'm simplifying them here for illustration purposes.

(1) $\quad f(x)=\text{MellinConvolve}[\sin(a\,x),\cos(b\,x),x,y]=\frac{1}{2}\pi\,_0\tilde{F}_1(;1;-a\,b\,y)$

(2) $\quad g(x)=\text{MellinConvolve}[\cos(a\,x),\cos(b\,x),x,y]=\frac{1}{2} \pi\,G_{0,4}^{2,0}\left(\frac{1}{16}a^2\,b^2\,y^2| \begin{array}{c} 0,0,\frac{1}{2},\frac{1}{2}\\ \end{array} \right)$

The following plot illustrates the $f(x)$ and $g(x)$ functions defined in (1) and (2) above with evaluation parameters $a=\frac{2\,\pi}{2}$ and $b=\frac{2\,\pi}{3}$. The $f(x)$ function is shown in blue and the $g(x)$ function is shown in orange. Note that $f(x)$ and $g(x)$ have the same amplitude but are orthogonal in phase analogous to the $sin(a\,x)$ and $cos(a\,x)$ functions, and that as $x$ increases the amplitude and frequency of $f(x)$ and $g(x)$ both decrease.

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Question: Is there a well developed theory with respect to expressing functions as infinite series of orthogonal hypergeometric functions (analogous to the way Fourier series are used to express functions as infinite series of $sin$ and/or $cos$ terms), and if so can someone provide me with a reference?

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  • $\begingroup$ Why are you messing up with the hypergeometric functions ? What is $MellinConvolve[sin(ax),cos(bx),x,y]$ ? (you should stop playing with mathematica) $\endgroup$ – reuns May 20 '17 at 17:57
  • $\begingroup$ What means "The following plot illustrates the function $S(x)−(x−12)$ in blue, a Fourier series representation of the error term in green, and a hypergeometric series represention of the error term in red." ? $\endgroup$ – reuns May 20 '17 at 18:00
  • $\begingroup$ @user1952009 I'm investigating Bessel functions of the first kind $J_v(x)$ because I've noticed there seems to be a relationship between $J_v(x)$ functions and Mellin convolution, and also because zeta zero terms associated with von Mangoldt's formula for the second Chebyshev function can be represented by Fourier-Bessel series. $\endgroup$ – Steven Clark May 22 '17 at 16:32
  • $\begingroup$ @user1952009 $S(x)=Floor[x]=x-SawtoothWave(x)=x-\left(\frac{1}{2}-\frac{1}{\pi}\sum_{k=1}^\infty\sin(2\,k\,\pi\,x)\right)$, so $S(x)-\left(x-\frac{1}{2}\right)=\frac{1}{\pi}\sum_{k=1}^\infty\sin(2\,k\,\pi\,x)$ which is represented in green (where the sum is over the first 4 harmonics). $\endgroup$ – Steven Clark May 22 '17 at 17:06
  • $\begingroup$ @user1952009 What I referred to as the Hypergeometric series (illustrated in red) is $MellinConvolve[\frac{1}{\pi}\sum_{k=1}^4\sin(2\,k\,\pi\,x)],\delta(x-1),x,y]$ where the Fourier series for $\delta(x-1)$ is used to evaluate this convolution. The result of this convolution is a series of $J_0$ terms (Bessel functions of the first kind), and since $MellinConvolve[f(x),\delta(x-1),x,y]=f(y)$, this is a $J_0$ series representation of $S(x)-\left(x-\frac{1}{2}\right)$. $\endgroup$ – Steven Clark May 22 '17 at 17:10
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I originally noticed the Fourier series representation of $\delta(x-1)$ can be used to rewrite $\sin$ and $\cos$ terms and hence any Fourier series $f(x)$ as a series of Bessel functions of the first kind via the Mellin convolution (1) below. For example, in the question above I illustrated the results of converting the Fourier series representation of the error term $\psi(x)-(x-\log(2\,\pi))$ of the second Chebyshev function into infinite series of Bessel $J_0(x)$ functions. Note Bessel functions are essentially special cases which can be represented by more general Hypergeometric functions.

(1) $\quad MellinConvolve\left[\delta(x-1),f(x),x,y\right]=\int_0^\infty \delta(x-1)\frac{f\left(\frac{y}{x}\right)}{x}\,dx=f(y)$

Since I originally posted this question, I've found some information about Fourier-Bessel series which differ somewhat from the approach illustrated in the question above, but which yield some interesting results nevertheless (see Fourier-Bessel Series Expansions of Zeta Zero Terms).

The following sites provide information about Fourier-Bessel series. The first link below is a bit confusing as the terminology changes from $A_r$ to $A_i$ in the middle of the explanation. The first link focuses on the expansion of a function in the interval $(0,1)$, whereas the second link covers the expansion of a function in the more general interval $(0,b)$. Even after taking into account the difference in symbols and the difference in intervals between the two links below, the second link below does not seem correct with respect to the denominator in the formula for the coefficient. I've been able to obtain convergent results using the coefficient formula defined in the first link below, but I've found the results to diverge when attempting to use the coefficient formula defined in the second link below. This could perhaps be an error or misinterpretation on my part.

Fourier-Bessel Series Mathworld

Fourier-Bessel Series Wikipedia

The second link above also briefly describes a second Fourier-Bessel series known as the Dini-series. I've noticed there seems to be a relationship between the Robin boundary condition $b\,f'(b)+c\,f(b)=0$ for $c=1$ and the following Mellin convolution.

(2) $\quad MellinConvolve\left[\delta'(x-1),f(x),x,b\right]=\int_0^\infty \delta'(x-1)\frac{f\left(\frac{b}{x}\right)}{x}\,dx=b\,f'(b)+f(b)$.

The Fourier-Bessel series is intended to represent the expansion of a function over a finite interval. The following links provide some information about the Hankel transform (also called the Fourier-Bessel transform) which is the counterpart of the Fourier-Bessel series over an infinite interval. The second link below also provides some information about an alternate definition of the Hankel transform.

Hankel Transform Mathworld

Hankel Transform Wikipedia

The following link provides some information on the $Y$ and $H$ transforms which are closely related to the Hankel transform (both involve Bessel functions).

$Y$ and $H$ Transforms Wikipedia

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