Further Examples

Abstract

The scope of applications of our theory is very wide since it applies to any function lying in the domain of the map Σ. In Chap. 10, we made a thorough study of some standard special functions. In Chap. 11, we defined and investigated new functions as principal indefinite sums of known functions. In the present chapter, we briefly discuss further examples that the reader may want to explore in more detail.

The scope of applications of our theory is very wide since it applies to any function lying in the domain of the map Σ. In Chap. 10, we made a thorough study of some standard special functions. In Chap. 11, we defined and investigated new functions as principal indefinite sums of known functions. In the present chapter, we briefly discuss further examples that the reader may want to explore in more detail.

12.1 The Multiple Gamma Functions

The multiple gamma functions introduced in Sect. 5.2 can also be studied through the sequence of functions G ₀, G ₁, …, defined by (see Srivastava and Choi [93, p. 56])

$$\displaystyle \begin{aligned} G_p(x) ~=~ \Gamma_p(x)^{(-1)^{p-1}},\qquad p\in\mathbb{N}. \end{aligned}$$

Equivalently, we have G ₀(x) = x and

$$\displaystyle \begin{aligned} \ln G_p(x) ~=~ \Sigma\ln G_{p-1}(x)\qquad \mbox{for all }p\in\mathbb{N}^*. \end{aligned}$$

Clearly, the function $\ln G_{p-1}(x)$ lies in $\mathcal {C}^{\infty }\cap \mathcal {D}^p\cap \mathcal {K}^{\infty }$ and we have $\deg (\ln \circ G_p)=p$. Moreover, this sequence of functions can naturally be extended to p = −1 by defining

$$\displaystyle \begin{aligned} G_{-1}(x) ~=~ 1+\frac{1}{x}{\,}. \end{aligned}$$

Just as for the gamma function and the Barnes G-function, we can derive the following asymptotic equivalence: for any a ≥ 0,

$$\displaystyle \begin{aligned} G_p(x+a) ~\sim ~ \prod_{j=0}^pG_{p-j}(x)^{{a\choose j}} \qquad \mbox{as }x\to\infty, \end{aligned}$$

with equality if a ∈{0, 1, …, p}. We also have the following product representation

$$\displaystyle \begin{aligned} G_p(x) ~=~ \frac{1}{G_{p-1}(x)}{\,} \prod_{k=1}^{\infty}\frac{G_{p-1}(k)}{G_{p-1}(x+k)}{\,}G_{p-2}(k)^x G_{p-3}(k)^{{x\choose 2}}{\,}\cdots {\,} G_{-1}(k)^{{x\choose p}} \end{aligned}$$

and the recurrence formula

$$\displaystyle \begin{aligned} \ln G_p(x) ~=~ -(x-1){\,}\sigma[D\ln\circ G_{p-1}]+\int_1^x\Sigma D\ln G_{p-1}(t){\,}dt. \end{aligned}$$

For example, one can show that

$$\displaystyle \begin{aligned}\displaystyle \ln G_3(x) ~=~ -\frac{1}{8}{\,}x(x-1)(2x-5)+\frac{1}{4}{\,}x(x-2)\ln(2\pi)+{\textstyle{{{x-1}\choose{2}}}}\ln\Gamma(x)\\\displaystyle -\frac{1}{2}(2x-3){\,}\psi_{-2}(x)+\psi_{-3}(x)-x\,\psi_{-3}(1). \end{aligned} $$

This latter formula can also be established using the characterization of G ₃ as a 3-convex solution to the equation $\Delta f(x)=\ln G_2(x)$.

12.2 The Regularized Incomplete Gamma Function

Consider the 2-variable function Q(x, s) =  Γ(x, s)∕ Γ(x) on $\mathbb {R}_+^2$, where Γ(x, s) is the upper incomplete gamma function. Thus defined, the function Q(x, s) satisfies the difference equation

$$\displaystyle \begin{aligned} Q(x+1,s)-Q(x,s) ~=~ \frac{e^{-s}s^x}{\Gamma(x+1)}{\,}. \end{aligned}$$

For any s > 0, we define the function $g_s\colon \mathbb {R}_+\to \mathbb {R}$ by

$$\displaystyle \begin{aligned} g_s(x) ~=~ \frac{e^{-s}s^x}{\Gamma(x+1)}{\,}. \end{aligned}$$

This function lies in $\mathcal {C}^{\infty }\cap \mathcal {D}^{-1}\cap \mathcal {K}^{\infty }$ and has the property that Σg _s(x) = Q(x, s) − e ^−s. We also note that the Eulerian form of Q(x, s) is

$$\displaystyle \begin{aligned} \begin{array}{rcl} Q(x,s) & =&\displaystyle 1-\sum_{k=0}^{\infty}g_s(x+k) ~=~ 1-\frac{e^{-s}s^x}{\Gamma(x+1)}\,\sum_{k=0}^{\infty}\frac{\Gamma(x+1)}{\Gamma(x+k+1)}{\,}s^k\\ & =&\displaystyle 1- \frac{e^{-s}s^x}{\Gamma(x+1)}\,\sum_{k=0}^{\infty}x^{\underline{-k}}{\,}s^k{\,}, \end{array} \end{aligned} $$

where $x^{ \underline {-k}}=\Gamma (x+1)/\Gamma (x+k+1)$ for any $k\in \mathbb {N}$.

12.3 The Error Function

Recall that the Gauss error function erf(x) is defined by the equation

$$\displaystyle \begin{aligned} \mathrm{erf}(x) ~=~ \frac{2}{\sqrt{\pi}}\,\int_0^xe^{-t^2}{\,}dt\qquad \mbox{for }x>0. \end{aligned}$$

To study this function, we could for instance work with the function g(x) =  Δerf(x). Instead, let us consider the function $g\colon \mathbb {R}_+\to \mathbb {R}$ defined by the equation

$$\displaystyle \begin{aligned} g(x) ~=~ \frac{2}{\sqrt{\pi}}{\,}e^{-x^2}\qquad \mbox{for }x>0. \end{aligned}$$

It clearly lies in $\mathcal {C}^{\infty }\cap \mathcal {D}^{-1}\cap \mathcal {K}^{\infty }$. Thus, the Eulerian form of Σg is given by the identity

$$\displaystyle \begin{aligned} \Sigma g(x) ~=~ \frac{2}{\sqrt{\pi}}\,\sum_{k=0}^{\infty}(e^{-(k+1)^2}-e^{-(k+x)^2}). \end{aligned}$$

The generalized Stirling formula yields the following limit

$$\displaystyle \begin{aligned} \mathrm{erf}(x) + \frac{2}{\sqrt{\pi}}\,\sum_{k=0}^{\infty}e^{-(k+x)^2} ~\to ~ 1\qquad \mbox{as }x\to\infty. \end{aligned}$$

Incidentally, the analogue of Legendre’s duplication formula provides the surprising identity

$$\displaystyle \begin{aligned} \sum_{k=0}^{\infty}(e^{-(k+1)^2}-e^{-(k+\frac{x}{2})^2}-e^{-(k+\frac{x+1}{2})^2}+e^{-(k+\frac{1}{2})^2} -e^{-(\frac{k+1}{2})^2}+e^{-(\frac{k+x}{2})^2}) ~=~ 0. \end{aligned}$$

12.4 The Exponential Integral

Recall that the exponential integral E ₁(x) is defined by the equation

$$\displaystyle \begin{aligned} E_1(x) ~=~ \int_x^{\infty}\frac{e^{-t}}{t}{\,}dt\qquad \mbox{for }x>0. \end{aligned}$$

Similarly to the previous example, let us consider the function $g\colon \mathbb {R}_+\to \mathbb {R}$ defined by the equation

$$\displaystyle \begin{aligned} g(x) ~=~ \frac{e^{-x}}{x}\qquad \mbox{for }x>0. \end{aligned}$$

It lies in $\mathcal {C}^{\infty }\cap \mathcal {D}^{-1}\cap \mathcal {K}^{\infty }$. Thus, the Eulerian form of Σg is given by the identity

$$\displaystyle \begin{aligned} \Sigma g(x) ~=~ \sum_{k=0}^{\infty}\left(\frac{e^{-(k+1)}}{k+1}-\frac{e^{-(k+x)}}{k+x}\right). \end{aligned}$$

The generalized Stirling formula easily provides the following convergence result

$$\displaystyle \begin{aligned} E_1(x) -\sum_{k=0}^{\infty}\frac{e^{-(k+x)}}{k+x} ~\to ~ 0\qquad \mbox{as }x\to\infty. \end{aligned}$$

Moreover, the analogue of Raabe’s formula is

$$\displaystyle \begin{aligned} \int_x^{x+1}\Sigma g(t){\,}dt ~=~ 1-\ln(e-1)-E_1(x),\qquad x>0. \end{aligned}$$

12.5 The Hyperfactorial Function

The hyperfactorial function (or K-function) is the function $K\colon \mathbb {R}_+\to \mathbb {R}_+$ defined by the equation $\ln K=\Sigma g$, where the function $g(x)=x\ln x$ lies in $\mathcal {C}^{\infty }\cap \mathcal {D}^2\cap \mathcal {K}^{\infty }$. Since we also have

$$\displaystyle \begin{aligned} g(x) ~=~ x+\Delta \psi_{-2}(x)-\psi_{-2}(1), \end{aligned}$$

we immediately derive (see also Example 8.21)

$$\displaystyle \begin{aligned} \ln K(x) ~=~ \Sigma g(x) ~=~ {\textstyle{{{x}\choose{2}}}}+\psi_{-2}(x)-x\,\psi_{-2}(1) ~=~ (x-1)\ln\Gamma(x)-\ln G(x). \end{aligned}$$

Actually, g also corresponds to the special case when (s, q) = (−1, 1) of the function g _s,q investigated in Sect. 10.8. Thus, we also have

$$\displaystyle \begin{aligned} \Sigma g(x) ~=~ \zeta'(-1,x)-\zeta'(-1), \end{aligned}$$

where $\zeta '(-1)=\frac {1}{12}-\ln A$. Finally, we note that the integer sequence n↦K(n) is the sequence A002109 in the OEIS [90].

12.6 The Hurwitz-Lerch Transcendent

The Hurwitz-Lerch transcendent Φ(z, s, a) is a generalization of the Hurwitz zeta function defined as an analytic continuation of the series

$$\displaystyle \begin{aligned} \sum_{k=0}^{\infty}z^k(a+k)^{-s} \end{aligned}$$

when |z| < 1 and $a\in \mathbb {C}\setminus (-\mathbb {N})$ (see, e.g., Srivastava and Choi [93, p. 194]). It satisfies the difference equation

$$\displaystyle \begin{aligned} \Phi(z,s,a+1)-z^{-1}\Phi(z,s,a) ~=~ -z^{-1}a^{-s}. \end{aligned}$$

It follows that the modified function

$$\displaystyle \begin{aligned} \overline{\Phi}(z,s,a) ~=~ -z^a\,\Phi(z,s,a) \end{aligned}$$

satisfies the difference equation

$$\displaystyle \begin{aligned} \overline{\Phi}(z,s,a+1)-\overline{\Phi}(z,s,a) ~=~ z^aa^{-s}. \end{aligned}$$

Thus, for certain real values of z and s, the restriction to $\mathbb {R}_+$ of the map $a\mapsto \overline {\Phi }(z,s,a)$ fits the assumptions of our theory. Its investigation is left to the reader.

12.7 The Bernoulli Polynomials

Recall that, for any $n\in \mathbb {N}$, the nth degree Bernoulli polynomial B _n(x) is defined by the equation

$$\displaystyle \begin{aligned} B_n(x) ~=~ \sum_{k=0}^n{\textstyle{{{n}\choose{k}}}}{\,}B_{n-k}{\,}x^k\qquad \mbox{for }x\in\mathbb{R}, \end{aligned}$$

where B _k is the kth Bernoulli number. This polynomial satisfies the difference equation

$$\displaystyle \begin{aligned} B_n(x+1)-B_n(x) ~=~ n{\,}x^{n-1}. \end{aligned}$$

Thus, the function $g_n\colon \mathbb {R}_+\to \mathbb {R}$ defined by the equation g _n(x) = n x ⁿ⁻¹ for x > 0 lies in $\mathcal {C}^{\infty }\cap \mathcal {D}^n\cap \mathcal {K}^{\infty }$ and has the property that

$$\displaystyle \begin{aligned} \Sigma g_n(x) ~=~ B_n(x)-B_n(1), \end{aligned}$$

that is, in view of (10.16)

$$\displaystyle \begin{aligned} \Sigma g_n(x) ~=~ n\zeta(1-n)-n\zeta(1-n,x){\,},\qquad n\in\mathbb{N}^*. \end{aligned}$$

Thus, the nth degree Bernoulli polynomial can be characterized as follows.

All solutions $f_n\colon \mathbb {R}_+\to \mathbb {R}$ to the equation f _n(x + 1) − f _n(x) = n x ⁿ⁻¹ that lie in $\mathcal {K}^n$ are of the form f _n(x) = c _n + B _n(x), where $c_n\in \mathbb {R}$.

Using the generalized Webster functional equation (Theorem 8.71), we can also easily characterize the nth degree Euler polynomial E _n(x), which is defined by the equation

$$\displaystyle \begin{aligned} E_n(x) ~=~ \frac{2^{n+1}}{n+1}\left(B_{n+1}\left(\frac{x+1}{2}\right)-B_{n+1}\left(\frac{x}{2}\right)\right). \end{aligned}$$

We then obtain the following statement.

All solutions $f_n\colon \mathbb {R}_+\to \mathbb {R}$ to the equation f _n(x + 1) + f _n(x) = 2 x ⁿ that lie in $\mathcal {K}^n$ are of the form f _n(x) = c _n + E _n(x), where $c_n\in \mathbb {R}$.

Finally, we also easily retrieve the multiplication formula:

$$\displaystyle \begin{aligned} \sum_{j=0}^{m-1}B_n\left(\frac{x+j}{m}\right) ~=~ \frac{1}{m^{n-1}}{\,}B_n(x){\,}\qquad x>0. \end{aligned}$$

12.8 The Bernoulli Polynomials of the Second Kind

For any $n\in \mathbb {N}$, the nth degree Bernoulli polynomial of the second kind is defined by the equation

$$\displaystyle \begin{aligned} \psi_n(x) ~=~ \int_x^{x+1}{\textstyle{{{t}\choose{n}}}}{\,}dt\qquad \mbox{for }x>0. \end{aligned}$$

In particular, we have ψ _n(0) = G _n. Also, these polynomials satisfy the difference equation

$$\displaystyle \begin{aligned} \psi_{n+1}(x+1)-\psi_{n+1}(x) ~=~ \psi_n(x). \end{aligned}$$

Thus, the function $g_n\colon \mathbb {R}_+\to \mathbb {R}$ defined by the equation g _n(x) = ψ _n(x) for x > 0 lies in $\mathcal {C}^{\infty }\cap \mathcal {D}^{n+1}\cap \mathcal {K}^{\infty }$ and has the property that

$$\displaystyle \begin{aligned} \Sigma g_n(x) ~=~ \psi_{n+1}(x)-\psi_{n+1}(1). \end{aligned}$$

Thus, the Bernoulli polynomials of the second kind can be characterized as follows.

All solutions $f_n\colon \mathbb {R}_+\to \mathbb {R}$ to the equation f _n(x + 1) − f _n(x) = ψ _n(x) that lie in $\mathcal {K}^{n+1}$ are of the form f _n(x) = c _n + ψ _n+1(x), where $c_n\in \mathbb {R}$.