Defining New Multiple $\log \Gamma$-Type Functions

Abstract

In the previous chapter, we tested our results on some multiple $\log \Gamma$-type functions that are well-known special functions. It is clear, however, that there are many other multiple $\log \Gamma$-type functions that are still to be introduced and investigated, simply as principal indefinite sums of standard functions.

In the previous chapter, we tested our results on some multiple $\log \Gamma$-type functions that are well-known special functions. It is clear, however, that there are many other multiple $\log \Gamma$-type functions that are still to be introduced and investigated, simply as principal indefinite sums of standard functions.

In this chapter, we introduce and investigate the following functions (we use the acronym PIS for “principal indefinite sum”)

• The PIS of the digamma function.

• The PIS of the Hurwitz zeta function.

• The PIS of the generating function for the Gregory coefficients.

The latter two examples are examined here in a broad way. A deeper investigation of these examples can be carried out simply by following all the steps and recipes given in Chap. 9.

11.1 The PIS of the Digamma Function

Let us see what our theory tells us when g(x) = ψ(x) is the digamma function. We first observe that g lies in $\mathcal {C}^{\infty }\cap \mathcal {D}^1\cap \mathcal {K}^{\infty }$.

Using summation by parts, we can easily see that

$$\displaystyle \begin{aligned} \Sigma\psi(x) ~=~ (x-1)(\psi(x)-1). \end{aligned}$$

Moreover, from the identity H _x−1 = ψ(x) + γ, we obtain immediately

$$\displaystyle \begin{aligned} \Sigma_x H_{x-1} ~=~ (x-1)(H_{x-1} -1). \end{aligned}$$

This example may seem very basic at first glance, but since H _x is the discrete analogue of the function $\ln x$, we expect an important analogy between Σψ(x) and $\Sigma \ln x=\ln \Gamma (x)$, at least in terms of asymptotic behaviors. Actually, the analogue of Burnside’s formula shows that the function

$$\displaystyle \begin{aligned} \ln\Gamma\left(x-\frac{1}{2}\right)+\frac{1}{2}(1-\ln(2\pi)) \end{aligned}$$

is a very good approximation of Σψ(x).

Interestingly, using (10.12) we can easily derive the following additional identity

$$\displaystyle \begin{aligned} \Sigma\psi(x) ~=~ \frac{1}{2}\left(1-\ln(2\pi)\right)+ D\ln G(x),\qquad x>0, \end{aligned}$$

where G is the Barnes G-function (see Sect. 10.5).

Project 11.1

Find a closed-form expression for the function Σ_x ψ ²(x). Using again summation by parts, we obtain

$$\displaystyle \begin{aligned} \Sigma_x\psi^2(x) ~=~ (x-1)\,\psi^2(x)-(2x-1)\,\psi(x)+2x-2-\gamma. \end{aligned}$$

We also note that the function ψ ²(x) lies in $\mathcal {C}^{\infty }\cap \mathcal {D}^1\cap \mathcal {K}^{\infty }$, just as does the function ψ(x). The investigation of this new function in the light of our results is left to the reader. $\lozenge$

ID Card

The following basic information about the functions ψ(x) and Σψ(x) follows trivially from the discussion above.

g(x)	Membership	deg g	Σg(x)
$\mathstrut ^{\mathstrut }_{\mathstrut } \psi (x)$	$\mathstrut ^{\mathstrut }_{\mathstrut }\mathcal {C}^{\infty }\cap \mathcal {D}^1\cap \mathcal {K}^{\infty }$	0	(x − 1)(ψ(x) − 1)

Analogue of Bohr-Mollerup’s Theorem

The function Σψ(x) can be characterized as follows.

All eventually convex or concave solutions $f\colon \mathbb {R}_+\to \mathbb {R}$ to the equation

$$\displaystyle \begin{aligned} f(x+1)-f(x) ~=~ \psi(x) \end{aligned}$$

are of the form f(x) = c +  Σψ(x), where $c\in \mathbb {R}$.

Extended ID Card

It is not difficult to see that

$$\displaystyle \begin{aligned} \sigma[g] ~=~ \int_0^1\Sigma\psi(t+1){\,}dt ~=~ \frac{1}{2}(1-\ln(2\pi)). \end{aligned}$$

Hence we have the values

• Alternative representations of σ[g]

  $$\displaystyle \begin{aligned} \begin{array}{rcl} \sigma[g] & =&\displaystyle -\frac{1}{2}\,\gamma -\sum_{k=1}^{\infty}\left(\ln k-\psi(k)-\frac{1}{2k}\right),\\ \sigma[g] & =&\displaystyle -\frac{1}{2}\,\gamma +\int_1^{\infty}\left(\{t\}-\frac{1}{2}\right)\psi_1(t){\,}dt,\\ \sigma[g] & =&\displaystyle \lim_{n\to\infty}\left(\left(n-\frac{1}{2}\right)\psi(n)-\ln\Gamma(n)-n+1\right). \end{array} \end{aligned} $$

• Alternative representations of γ[g]

  $$\displaystyle \begin{aligned} \begin{array}{rcl} \sigma[g] & =&\displaystyle \int_1^{\infty}\left(\psi(\lfloor t\rfloor)-\psi(t)+\frac{1}{2\lfloor t\rfloor}\right)dt,\\ \sigma[g] & =&\displaystyle \int_1^{\infty}\left(\psi(\lfloor t\rfloor)-\psi(t)+\frac{\{t\}}{\lfloor t\rfloor}\right)dt. \end{array} \end{aligned} $$

• Generalized Binet’s function. For any $q\in \mathbb {N}^*$ and any x > 0,

  $$\displaystyle \begin{aligned} \begin{array}{rcl} J^{q+1}[\Sigma\psi](x) & =&\displaystyle \Sigma\psi(x)-\frac{1}{2}(1-\ln(2\pi))-\ln\Gamma(x)+\frac{1}{2}\,\psi(x)\\ & &\displaystyle +\sum_{j=0}^{q-2}G_{j+2}(-1)^j\,\mathrm{B}(j+1,x), \end{array} \end{aligned} $$

  where (x, y)↦B(x, y) is the beta function.

• Analogue of Raabe’s formula

  $$\displaystyle \begin{aligned} \int_x^{x+1}\Sigma\psi(t){\,}dt ~=~ \frac{1}{2}(1-\ln(2\pi))+\ln\Gamma(x),\qquad x>0. \end{aligned}$$

• Alternative characterization. The function f =  Σψ is the unique solution lying in $\mathcal {C}^0\cap \mathcal {K}^1$ to the equation

  $$\displaystyle \begin{aligned} \int_x^{x+1}f(t){\,}dt ~=~ \frac{1}{2}(1-\ln(2\pi))+\ln\Gamma(x),\qquad x>0. \end{aligned}$$

$\overline {\sigma }[g]$	σ[g]	γ[g]
$\mathstrut ^{\mathstrut }_{\mathstrut } \infty$	$\frac {1}{2}(1-\ln (2\pi ))$	$\frac {1}{2}(1-\ln (2\pi )+\gamma )$

Inequalities

The following inequalities hold for any x > 0, any a ≥ 0, and any $n\in \mathbb {N}^*$.

• Symmetrized generalized Wendel’s inequality (equality if a ∈{0, 1})

  $$\displaystyle \begin{aligned} \begin{array}{rcl} |\Sigma\psi(x+a)-\Sigma\psi(x)-a\psi(x)| & \leq &\displaystyle |a-1|{\,}|\psi(x+a)-\psi(x)|\\ & \leq &\displaystyle \lceil a\rceil\,\frac{|a-1|}{x}{\,}. \end{array} \end{aligned} $$

• Symmetrized generalized Wendel’s inequality (discrete version)

  $$\displaystyle \begin{aligned} |\Sigma\psi(x)-f^1_n[\psi](x)| ~\leq ~ |x-1|{\,}|\psi(n+x)-\psi(n)| ~\leq ~ \lceil x\rceil\,\frac{|x-1|}{n}{\,}, \end{aligned}$$

  where

  $$\displaystyle \begin{aligned} f^1_n[\psi](x) ~=~ (n+x-1)(\psi(n)-\psi(x+n))+(x-1)\,\psi(x)+1. \end{aligned}$$

• Symmetrized Stirling’s formula-based inequalities

  $$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle {\left|\Sigma\psi\left(x+\frac{1}{2}\right)-\frac{1}{2}(1-\ln(2\pi))-\ln\Gamma(x)\right|}\\ & \leq &\displaystyle \left|\Sigma\psi(x)-\frac{1}{2}(1-\ln(2\pi))-\ln\Gamma(x)+\frac{1}{2}\,\psi(x)\right|\\ & \leq &\displaystyle x\ln x-\ln\Gamma(x)-\frac{1}{2}\,\psi(x)-x+\frac{1}{2}\ln(2\pi) ~\leq ~ \frac{1}{2x}{\,}. \end{array} \end{aligned} $$

• Generalized Gautschi’s inequality

  $$\displaystyle \begin{aligned} \begin{array}{rcl} (a-\lceil a\rceil)\,\psi(x+\lceil a\rceil) & \leq &\displaystyle (a-\lceil a\rceil){\,}(\Sigma\psi)'(x+\lceil a\rceil)\\ & \leq &\displaystyle (\Sigma\psi)(x+a)-(\Sigma\psi)(x+\lceil a\rceil)\\ & \leq &\displaystyle (a-\lceil a\rceil)\,\psi(x+\lfloor a\rfloor). \end{array} \end{aligned} $$

Generalized Stirling’s and Related Formulas

For any a ≥ 0, we have the following limits and asymptotic equivalence as x →∞,

$$\displaystyle \begin{aligned} \Sigma\psi(x+a)-\Sigma\psi(x)-a\psi(x) ~\to ~0,\qquad \Sigma\psi(x+a) ~\sim ~ \ln\Gamma(x), \end{aligned}$$

$$\displaystyle \begin{aligned} \Sigma\psi(x)-\ln\Gamma(x)+\frac{1}{2}\,\psi(x) ~\to ~ \frac{1}{2}(1-\ln(2\pi)), \end{aligned}$$

$$\displaystyle \begin{aligned} \Sigma\psi(x)-\ln\Gamma\left(x-\frac{1}{2}\right) ~\to ~ \frac{1}{2}(1-\ln(2\pi)). \end{aligned}$$

Asymptotic Expansions

For any $q\in \mathbb {N}^*$ we have the following expansion as x →∞

$$\displaystyle \begin{aligned} \Sigma\psi(x) ~=~ \frac{1}{2}(1-\ln(2\pi))+\sum_{k=0}^q\frac{B_k}{k!}\,\psi_{k-1}(x)+O(\psi_q(x)). \end{aligned}$$

Setting q = 3 for instance, we get

$$\displaystyle \begin{aligned} \Sigma\psi(x) ~=~ \frac{1}{2}(1-\ln(2\pi))+\ln\Gamma(x)-\frac{1}{2}\,\psi(x)+\frac{1}{12}\,\psi_1(x)+O(x^{-3}). \end{aligned}$$

Generalized Liu’s Formula

For any x > 0, we have

$$\displaystyle \begin{aligned} \Sigma\psi(x) ~=~ \frac{1}{2}(1-\ln(2\pi))+\ln\Gamma(x)-\frac{1}{2}\,\psi(x)-\int_0^{\infty}\left(\{t\}-\frac{1}{2}\right)\psi_1(x+t){\,}dt. \end{aligned}$$

Limit and Series Representations

Let us briefly examine the main limit and series representations of Σψ(x). The additional representations obtained by differentiation and integration are left to the reader.

• Eulerian and Weierstrassian forms. We have

  $$\displaystyle \begin{aligned} \Sigma\psi(x) ~=~ -\gamma x-\psi(x)-\sum_{k=1}^{\infty}\left(\psi(x+k)-\psi(k)-\frac{x}{k}\right), \end{aligned}$$
  $$\displaystyle \begin{aligned} \Sigma\psi(x) ~=~ -(1+\gamma) x-\psi(x)-\sum_{k=1}^{\infty}\left(\psi(x+k)-\psi(k)-x\,\psi_1(k)\right). \end{aligned}$$

• Analogue of Gauss’ limit. We have

  $$\displaystyle \begin{aligned} \Sigma\psi(x) ~=~ (x-1)\,\psi(x)+1 + \lim_{n\to\infty}(n+x-1)(\psi(n)-\psi(x+n)). \end{aligned}$$

Gregory’s Formula-Based Series Representation

For any x > 0 we have

$$\displaystyle \begin{aligned} \Sigma\psi(x) ~=~ \frac{1}{2}(1-\ln(2\pi))+\ln\Gamma(x)-\frac{1}{2}\,\psi(x) + \sum_{n=0}^{\infty}|G_{n+2}|\,\mathrm{B}(n+1,x){\,}. \end{aligned}$$

Setting x = 1 in this identity yields the following analogue of Fontana-Mascheroni’s series

$$\displaystyle \begin{aligned} \sum_{n=2}^{\infty}\frac{|G_n|}{n-1} ~=~ -\frac{1}{2}+\frac{1}{2}\ln(2\pi)-\frac{1}{2}\,\gamma{\,}, \end{aligned}$$

and the right-hand value is precisely the generalized Euler constant γ[ψ] associated with the digamma function. We also observe that this latter identity was obtained by Kowalenko [52, p. 431].

Analogue of Gauss’ Multiplication Formula

Since we do not have any simple expression for the function $\Sigma _x\psi (\frac {x}{m})$, it seems difficult to find a usable multiplication formula here. We had the same difficulty in the investigation of the Barnes G-function (see Sect. 10.5). However, we can use Proposition 8.30 to derive the following convergence result. For any $m\in \mathbb {N}^*$ we have

$$\displaystyle \begin{aligned} \sum_{j=0}^{m-1}\Sigma\psi\left(\frac{x+j}{m}\right)-m\,\ln\Gamma\left(\frac{x}{m}\right) +\frac{1}{2}\,\psi\left(\frac{x}{m}\right) ~\to ~ \frac{m}{2}{\,}(1-\ln(2\pi))\quad \mbox{as }x\to\infty. \end{aligned}$$

Analogue of Wallis’s Product Formula

The following analogue of Wallis’s formula was already found in Project 10.1

$$\displaystyle \begin{aligned} \lim_{n\to\infty}\left(-\ln(4n)+2\sum_{k=1}^{2n}(-1)^k\psi(k)\right) ~=~ \gamma{\,}. \end{aligned}$$

Generalized Webster’s Functional Equation

For any $m\in \mathbb {N}^*$, there is a unique eventually monotone solution $f\colon \mathbb {R}_+\to \mathbb {R}$ to the equation

$$\displaystyle \begin{aligned} \sum_{j=0}^{m-1}f\left(x+\frac{j}{m}\right) ~=~ \psi(x) \end{aligned}$$

namely

$$\displaystyle \begin{aligned} f(x) ~=~ \Sigma\psi\left(x+\frac{1}{m}\right)-\Sigma\psi(x). \end{aligned}$$

Analogue of Euler’s Series Representation of γ

We have ( Σψ)′(1) = −1 − γ and

$$\displaystyle \begin{aligned} (\Sigma\psi)^{(k)}(1) ~=~ k\,\psi_{k-1}(1) ~=~ (-1)^k k!\,\zeta(k),\qquad k\geq 2. \end{aligned}$$

The Taylor series expansion of Σψ(x + 1) about x = 0 is

$$\displaystyle \begin{aligned} \Sigma\psi(x+1) ~=~ (-1-\gamma)x+\sum_{k=2}^{\infty}\zeta(k)(-x)^k,\qquad |x|<1. \end{aligned}$$

Integrating both sides of this equation on (0, 1), we obtain

$$\displaystyle \begin{aligned} \sum_{k=2}^{\infty}(-1)^k\frac{\zeta(k)}{k+1} ~=~ 1+\frac{1}{2}(\gamma -\ln(2\pi)). \end{aligned}$$

Analogue of the Reflection Formula

For any $x\in \mathbb {R}\setminus \mathbb {Z}$, we have

$$\displaystyle \begin{aligned} \Sigma\psi(1+x)+\Sigma\psi(1-x) ~=~ 1-\pi x\cot{}(\pi x). \end{aligned}$$

11.2 The PIS of the Hurwitz Zeta Function

In this section we apply our theory to investigate the function

for any fixed $s\in \mathbb {R}\setminus \{1\}$.

Using summation by parts, we observe that if s ≠ 2 we have

$$\displaystyle \begin{aligned} \zeta_2(s,x) ~=~ (x-1)\,\zeta(s,x)-\zeta(s-1,x)+\zeta(s-1). \end{aligned}$$

If s = 2, then

$$\displaystyle \begin{aligned} \zeta_2(2,x) ~=~ \Sigma_x\psi_1(x) ~=~ (x-1)\,\psi_1(x)+\psi(x)+\gamma. \end{aligned}$$

To keep this investigation simple, here we focus on some selected results only and we restrict ourselves to the case when s > 2, for which the sequence n↦ζ(s, n) is summable. In this case, by (6.23) we obtain immediately the following surprising identity (see also Paris [83])

$$\displaystyle \begin{aligned} \sum_{k=1}^{\infty}\zeta(s,k) ~=~ \zeta(s-1). \end{aligned}$$

We also have

$$\displaystyle \begin{aligned} \int_1^{\infty}\zeta(s,t){\,}dt ~=~ \frac{\zeta(s-1)}{s-1}{\,}. \end{aligned}$$

ID Card

We can easily summarize the basic information as follows:

g s(x)	Membership	deg g s	Σg s(x)
$\mathstrut ^{\mathstrut }_{\mathstrut } \zeta (s,x)$	$\mathstrut ^{\mathstrut }_{\mathstrut }\mathcal {C}^{\infty }\cap \mathcal {D}^{-1}\cap \mathcal {K}^{\infty }$	− 1	ζ 2(s, x)

Analogue of Bohr-Mollerup’s Theorem

The function ζ ₂(s, x) can be characterized as follows.

All eventually monotone solutions $f_s\colon \mathbb {R}_+\to \mathbb {R}$ to the equation

$$\displaystyle \begin{aligned} f_s(x+1)-f_s(x) ~=~ \zeta(s,x) \end{aligned}$$

are of the form f _s(x) = c _s + ζ ₂(s, x), where $c_s\in \mathbb {R}$.

Extended ID Card

We immediately have

$$\displaystyle \begin{aligned} \sigma[g_s] ~=~ \sum_{k=1}^{\infty}\zeta(s,k)-\int_1^{\infty}\zeta(s,t){\,}dt ~=~ \frac{s-2}{s-1}\,\zeta(s-1). \end{aligned}$$

Hence we have the values

• Alternative representations of σ[g _s] = γ[g _s]

  $$\displaystyle \begin{aligned} \begin{array}{rcl} \sigma[g_s] & =&\displaystyle \int_0^1\zeta_2(s,t+1){\,}dt ~=~ \int_1^{\infty}\left(\zeta(s,\lfloor t\rfloor)-\zeta(s,t)\right){\,}dt{\,},\\ \sigma[g_s] & =&\displaystyle \frac{1}{2}\,\zeta(s)+s\,\int_1^{\infty}\left(\frac{1}{2}-\{t\}\right)\zeta(s+1,t){\,}dt. \end{array} \end{aligned} $$

• Analogue of Raabe’s formula

  $$\displaystyle \begin{aligned} \int_x^{x+1}\zeta_2(s,t){\,}dt ~=~ \zeta(s-1)-\frac{\zeta(s-1,x)}{s-1}{\,},\qquad x>0. \end{aligned}$$

$\overline {\sigma }[g_s]$	σ[g s]	γ[g s]
$\mathstrut ^{\mathstrut }_{\mathstrut } \infty$	$\frac {s-2}{s-1}\,\zeta (s-1)$	γ[g s] = σ[g s]

Inequalities and Asymptotic Analysis

For any a ≥ 0 and any x > 0, we have

$$\displaystyle \begin{aligned} \begin{array}{rcl} \left|\zeta_2(s,x+a)-\zeta_2(s,x)\right| & \leq &\displaystyle \lceil a\rceil{\,}\zeta(s,x){\,},\\ \left|\zeta_2(s,x)-\zeta(s-1)+\frac{\zeta(s-1,x)}{s-1}\right| & \leq &\displaystyle \zeta(s,x). \end{array} \end{aligned} $$

In particular, we have

$$\displaystyle \begin{aligned} \zeta_2(s,x) ~\to ~\zeta(s-1)\qquad \mbox{as }x\to\infty. \end{aligned}$$

Generalized Liu’s Formula

For any x > 0 we have

$$\displaystyle \begin{aligned} \begin{array}{rcl} \zeta_2(s,x) & =&\displaystyle \zeta(s-1)-\frac{\zeta(s-1,x)}{s-1}-\frac{1}{2}\,\zeta(s,x)\\ & &\displaystyle +s\,\int_0^{\infty}\left(\{t\}-\frac{1}{2}\right)\zeta(s+1,x+t){\,}dt. \end{array} \end{aligned} $$

Eulerian and Weierstrassian Forms

For any x > 0, we have

$$\displaystyle \begin{aligned} \zeta_2(s,x) ~=~ \zeta(s-1)-\sum_{k=0}^{\infty}\zeta(s,x+k). \end{aligned}$$

and this series converges uniformly on $\mathbb {R}_+$ and can be integrated and differentiated term by term.

Gregory’s Formula-Based Series Representation

For any x > 0 we have

$$\displaystyle \begin{aligned} \begin{array}{rcl} \zeta_2(s,x) & =&\displaystyle \zeta(s-1)-\frac{\zeta(s-1,x)}{s-1}-\sum_{n=0}^{\infty}G_{n+1}\,\Delta_x^n\zeta(s,x)\\ & =&\displaystyle \zeta(s-1)-\frac{\zeta(s-1,x)}{s-1}-\sum_{n=0}^{\infty}|G_{n+1}|\,\sum_{k=0}^n(-1)^k{\textstyle{{{n}\choose{k}}}}\zeta(s,x+k){\,}. \end{array} \end{aligned} $$

Setting x = 1 in this identity yields the analogue of Fontana-Mascheroni series

$$\displaystyle \begin{aligned} \sum_{n=0}^{\infty}|G_{n+1}|\,\sum_{k=0}^n(-1)^k{\textstyle{{{n}\choose{k}}}}\zeta(s,k+1) ~=~ \frac{s-2}{s-1}\,\zeta(s-1){\,}. \end{aligned}$$

Analogue of Wallis’s Product Formula

The analogue of Wallis’s formula is

$$\displaystyle \begin{aligned} \begin{array}{rcl} \sum_{k=1}^{\infty}(-1)^{k-1}\zeta(s,k) & =&\displaystyle (2-2^{1-s})\zeta(s)+(1-2^{1-s})\zeta(s-1)\\ & &\displaystyle -2^{1-s}\,\sum_{k=0}^{\infty}\zeta\left(s,k+\frac{1}{2}\right). \end{array} \end{aligned} $$

This formula is actually obtained by combining Proposition 6.7 with the duplication formula for the Hurwitz zeta function

$$\displaystyle \begin{aligned} 2\,\zeta(s,2x) ~=~ 2^{1-s}\zeta(s,x)+2^{1-s}\zeta\left(s,x+\frac{1}{2}\right). \end{aligned}$$

On the other hand we also have (see Paris [83])

$$\displaystyle \begin{aligned} \sum_{k=1}^{\infty}(-1)^{k-1}\zeta(s,k) ~=~ (1-2^{-s})\,\zeta(s),\qquad s>1. \end{aligned}$$

Combining this formula with the analogue of Wallis’s formula, we derive the following identity

$$\displaystyle \begin{aligned} \sum_{k=0}^{\infty}\zeta\left(s,k+\frac{1}{2}\right) ~=~ (2^{s-1}-2^{-1})\,\zeta(s)+(2^{s-1}-1)\,\zeta(s-1). \end{aligned}$$

Taylor Series Expansion

We have

$$\displaystyle \begin{aligned} (\Sigma g_s)^{(k)}(1) ~=~ -k!\,{\textstyle{{{-s}\choose{k}}}}\,\zeta(s+k-1),\qquad k\in\mathbb{N}^*. \end{aligned}$$

The Taylor series expansion of ζ ₂(s, x + 1) about x = 0 is

$$\displaystyle \begin{aligned} \zeta_2(s,x+1) ~=~ -\sum_{k=1}^{\infty}{\textstyle{{{-s}\choose{k}}}}\,\zeta(s+k-1){\,}x^k{\,},\qquad |x|<1. \end{aligned}$$

11.3 The PIS of the Generating Function for the Gregory Coefficients

Let us investigate the function Σh _p for any $p\in \mathbb {N}^*$, where $h_p\colon \mathbb {R}_+\to \mathbb {R}$ is defined by the equation

$$\displaystyle \begin{aligned} h_p(x) ~=~ \frac{x^p}{\ln(x+1)} ~=~ x^p\,\mathrm{li}'(x+1)\qquad \mbox{for }x>0 \end{aligned}$$

and li(x) is the logarithmic integral function defined for all positive real numbers x ≠ 1 by the integral

$$\displaystyle \begin{aligned} \mathrm{li}(x) ~=~ \int_0^x\frac{1}{\ln t}{\,}dt{\,}. \end{aligned}$$

Incidentally, when p = 1, this function reduces to the ordinary generating function for the sequence n↦G _n. That is,

$$\displaystyle \begin{aligned} h_1(x) ~=~ \sum_{n=0}^{\infty}G_n{\,}x^n,\qquad |x|<1. \end{aligned}$$

More generally, h _p(x) = x ^p−1 h ₁(x) is the ordinary generating function for the right-shifted sequence n↦G _n−p+1, that is the sequence

$$\displaystyle \begin{aligned} 0,\ldots,0,G_0,G_1,G_2,\ldots \end{aligned}$$

with p − 1 leading 0’s.

We also note that the function h _p has the following integral representation

$$\displaystyle \begin{aligned} h_p(x) ~=~ x^{p-1}\int_0^1(x+1)^s{\,}ds. \end{aligned}$$

This latter representation actually suggests introducing, for any $p\in \mathbb {N}^*$, the function $g_p\colon \mathbb {R}_+\to \mathbb {R}$ defined by the equation

$$\displaystyle \begin{aligned} g_p(x) ~=~ \int_0^1(x+1)^{s+p-1}{\,}ds ~=~ \frac{x(x+1)^{p-1}}{\ln(x+1)}\qquad \mbox{for }x>0. \end{aligned}$$

The conversion formulas between the $h_p^{\prime }s$ and the $g_p^{\prime }s$ are simply given by the following equations

$$\displaystyle \begin{aligned} \begin{array}{rcl} g_p(x) & =&\displaystyle \sum_{k=1}^p{\textstyle{{{p-1}\choose{k-1}}}}{\,}h_k(x){\,},\\ h_p(x) & =&\displaystyle \sum_{k=1}^p{\,}(-1)^{p-k}{\textstyle{{{p-1}\choose{k-1}}}}{\,}g_k(x){\,}. \end{array} \end{aligned} $$

In particular, we have g ₁ = h ₁.

Since the function g _p has a nicer integral form than h _p, for the sake of simplicity we will investigate the function Σg _p for any $p\in \mathbb {N}^*$. By Proposition 5.7, the function Σh _p can then be obtained by applying the operator Σ to both sides of the second conversion formula above.

Remark 11.2

We observe that the function g _p is also the ordinary generating function for the sequence n↦ψ _n(p − 1), where ψ _n is the nth degree Bernoulli polynomial of the second kind (see Sect. 12.8). $\lozenge$

ID Card

It is not difficult to see that both g _p and h _p lie in $\mathcal {C}^{\infty }\cap \mathcal {D}^p\cap \mathcal {K}^{\infty }$ and hence also in $\mathcal {K}^p$. We also have deg g _p =deg h _p = p − 1.

From the integral form of g _p above, we can easily derive the following explicit form of Σg _p (after replacing 1 − s with s in the integral)

$$\displaystyle \begin{aligned} \Sigma g_p(x) ~=~ \int_0^1\zeta(s-p,2){\,}ds-\int_0^1\zeta(s-p,x+1){\,}ds, \end{aligned}$$

that is,

$$\displaystyle \begin{aligned} \Sigma g_p(x) ~=~ \tau_p-\int_0^1\zeta(s-p,x+1){\,}ds{\,}, \end{aligned}$$

with

$$\displaystyle \begin{aligned} \tau_p ~=~ -1+\int_0^1\zeta(s-p){\,}ds{\,}, \end{aligned}$$

where ζ(s, x) is the Hurwitz zeta function.

Remark 11.3

For any integer n ≥ 2, the harmonic number function of order n is defined on (−1, ∞) by

$$\displaystyle \begin{aligned} x ~\mapsto ~ H^{(n)}_x ~=~ \zeta(n)-\zeta(n,x+1), \end{aligned}$$

see, e.g., Srivastava and Choi [93, p. 266]. Extending this definition to noninteger orders by writing

$$\displaystyle \begin{aligned} H_x^{(s)} ~=~ \zeta(s)-\zeta(s,x+1),\qquad s\in\mathbb{R}\setminus\{1\}, \end{aligned}$$

we obtain the following very compact integral representation

$$\displaystyle \begin{aligned} \Sigma g_p(x) ~=~ -1+\int_0^1H_x^{(s-p)}{\,}ds{\,},\qquad x>0. \end{aligned}$$

$\lozenge$

Analogue of Bohr-Mollerup’s Theorem

Thus defined, Σh _p is a $\log \Gamma _p$-type function that lies in $\mathcal {C}^{\infty }\cap \mathcal {D}^{p+1}\cap \mathcal {K}^{\infty }$. This function can be characterized as follows.

All solutions $f\colon \mathbb {R}_+\to \mathbb {R}$ to the equation Δf = h _p that lie in $\mathcal {K}^p$ are of the form

$$\displaystyle \begin{aligned} f(x) ~=~ c_p + \sum_{k=1}^p{\,}(-1)^{p-k}{\textstyle{{{p-1}\choose{k-1}}}}{\,}\Sigma g_k(x){\,}, \end{aligned}$$

where $c_p\in \mathbb {R}$.

Extended ID Card

Let us compute the asymptotic constant associated with the function g _p. We have

$$\displaystyle \begin{aligned} \begin{array}{rcl} \sigma[g_p] & =&\displaystyle \int_0^1\Sigma g_p(t+1){\,}dt ~=~ \tau_p-\int_0^1\!\int_0^1\zeta(s-p,t+2){\,}dt{\,}ds\\ & =&\displaystyle \tau_p +\int_0^1\frac{2^{s+p}}{s+p}{\,}ds{\,}. \end{array} \end{aligned} $$

Using the change of variable u = 2^s+p, we finally obtain

$$\displaystyle \begin{aligned} \sigma[g_p] ~=~ \tau_p + \int_{2^p}^{2^{p+1}}\frac{1}{\ln t}{\,}dt ~=~ \tau_p +\mathrm{li}(2^{p+1})-\mathrm{li}(2^p). \end{aligned}$$

Now, we have

$$\displaystyle \begin{aligned} \begin{array}{rcl} \int_1^xg_p(t){\,}dt & =&\displaystyle \int_0^1\frac{(x+1)^{s+p}-2^{s+p}}{s+p}{\,}ds\\ & =&\displaystyle \mathrm{li}((x+1)^{p+1})-\mathrm{li}((x+1)^p)-\mathrm{li}(2^{p+1})+\mathrm{li}(2^p) \end{array} \end{aligned} $$

and hence the analogue of Raabe’s formula is

$$\displaystyle \begin{aligned} \int_x^{x+1}\Sigma g_p(t){\,}dt ~=~ \tau_p +\mathrm{li}((x+1)^{p+1})-\mathrm{li}((x+1)^p){\,},\qquad x>0. \end{aligned}$$

Generalized Stirling’s and Related Formulas When p = 1

For any a ≥ 0, we have the following limits and asymptotic equivalence as x →∞,

$$\displaystyle \begin{aligned} \Sigma g_1(x+a)-\Sigma g_1(x)-a{\,}\frac{x}{\ln(x+1)} ~\to ~ 0, \end{aligned}$$

$$\displaystyle \begin{aligned} \Sigma g_1(x) -\mathrm{li}((x+1)^2)+\mathrm{li}(x+1)+\frac{x}{2\ln(x+1)}~\to ~ \tau_1{\,}, \end{aligned}$$

$$\displaystyle \begin{aligned} \Sigma g_1(x+a) ~\sim ~ \mathrm{li}((x+1)^2)-\mathrm{li}(x+1). \end{aligned}$$

Upon differentiation,

$$\displaystyle \begin{aligned} D\Sigma g_1(x)-\frac{x-\frac{1}{2}}{\ln(x+1)} ~\to ~0,\qquad D^{k+1}\Sigma g_1(x) ~\to ~0,\quad k\in\mathbb{N}^*, \end{aligned}$$

$$\displaystyle \begin{aligned} D\Sigma g_1(x+a) ~\sim ~ \frac{x}{\ln(x+1)}{\,}, \end{aligned}$$

where

$$\displaystyle \begin{aligned} D\Sigma g_1(x) ~=~ \int_0^1(s-1)\,\zeta(s,x+1){\,}ds. \end{aligned}$$

Limit and Series Representations When p = 1

The Eulerian and Weierstrassian forms are

$$\displaystyle \begin{aligned} \Sigma g_1(x) ~=~ -g_1(x)+x{\,}g_1(1)-\sum_{k=1}^{\infty}\left(g_1(x+k)-g(k)-x\,\Delta_kg_1(k)\right) \end{aligned}$$

and

$$\displaystyle \begin{aligned} \Sigma g_1(x) ~=~ -g_1(x)+x{\,}D\Sigma g_1(1)-\sum_{k=1}^{\infty}\left(g_1(x+k)-g(k)-x{\,}g^{\prime}_1(k)\right), \end{aligned}$$

where

$$\displaystyle \begin{aligned} D\Sigma g_1(1) ~=~ \int_0^1(s-1)\,\zeta(s,2){\,}ds ~=~ \frac{1}{2}-\int_0^1s\,\zeta(1-s){\,}ds. \end{aligned}$$

Gregory’s Formula-Based Series Representation When p = 1

Proposition 8.11 provides the following series representation: for any x > 0 we have

$$\displaystyle \begin{aligned} \begin{array}{rcl} \Sigma g_1(x) & =&\displaystyle \tau_1 +\mathrm{li}((x+1)^2)-\mathrm{li}(x+1)-\sum_{n=0}^{\infty}G_{n+1}\,\Delta^ng(x)\\ & =&\displaystyle \tau_1 +\mathrm{li}((x+1)^2)-\mathrm{li}(x+1)-\sum_{n=0}^{\infty}|G_{n+1}|\,\sum_{k=0}^n(-1)^k{\textstyle{{{n}\choose{k}}}}\,\frac{x+k}{\ln(x+k+1)}{\,}. \end{array} \end{aligned} $$

Setting x = 1 in this identity, we obtain the following analogue of Fontana-Mascheroni’s series

$$\displaystyle \begin{aligned} \sigma[g_1] ~=~ \tau_1 +\mathrm{li}(4)-\mathrm{li}(2) ~=~ \sum_{n=0}^{\infty}|G_{n+1}|\,\sum_{k=0}^n(-1)^k{\textstyle{{{n}\choose{k}}}}\,\frac{k+1}{\ln(k+2)}{\,}. \end{aligned}$$

Analogue of Gauss’ Multiplication Formula

For any $m\in \mathbb {N}^*$ and any x > 0, we have

$$\displaystyle \begin{aligned} \sum_{j=0}^{m-1}\Sigma g_p\left(x+\frac{j}{m}\right) ~=~ m{\,}\tau_p-\int_0^1\sum_{j=0}^{m-1}\zeta\left(s-p,x+1+\frac{j}{m}\right)ds. \end{aligned}$$

Using the multiplication formula for the Hurwitz zeta function, we then obtain the following analogue of Gauss’ multiplication formula

$$\displaystyle \begin{aligned} \sum_{j=0}^{m-1}\Sigma g_p\left(x+\frac{j}{m}\right) ~=~ m{\,}\tau_p-\int_0^1m^{s-p}\,\zeta(s-p,mx+m){\,}ds. \end{aligned}$$

Now, using (8.15) we obtain

$$\displaystyle \begin{aligned} \begin{array}{rcl} \Sigma_x{\,} g_p\left(\frac{x}{m}\right) & =&\displaystyle \sum_{j=0}^{m-1}\Sigma g_p\left(\frac{x+j}{m}\right) - \sum_{j=1}^m\Sigma g_p\left(\frac{j}{m}\right)\\ & =&\displaystyle \int_0^1m^{s-p}\left(\zeta(s-p,m+1)-\zeta(s-p,x+m)\right)ds. \end{array} \end{aligned} $$

Corollary 8.33 then tells us that the sequences

$$\displaystyle \begin{aligned} m ~\mapsto ~ \int_0^1m^{s-p-1}\left(\zeta(s-p,2m)-\zeta(s-p,mx+m)\right)ds \end{aligned}$$

and

$$\displaystyle \begin{aligned} m ~\mapsto ~ \int_0^1m^{s-p-1}\left(\zeta(s-p,m+1)-\zeta(s-p,mx+m)\right)ds \end{aligned}$$

converge to the integrals

$$\displaystyle \begin{aligned} \int_1^x g_p(t){\,}dt \qquad \mbox{and}\qquad \int_0^x g_p(t){\,}dt{\,}, \end{aligned}$$

respectively.