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.
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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”)
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The PIS of the digamma function.
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The PIS of the Hurwitz zeta function.
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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
Moreover, from the identity H x−1 = ψ(x) + γ, we obtain immediately
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
is a very good approximation of Σψ(x).
Interestingly, using (10.12) we can easily derive the following additional identity
where G is the Barnes G-function (see Sect. 10.5).
Project 11.1
Find a closed-form expression for the function Σx ψ 2(x). Using again summation by parts, we obtain
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
Hence we have the values
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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.
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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 \(n\in \mathbb {N}^*\).
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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 →∞,
Asymptotic Expansions
For any \(q\in \mathbb {N}^*\) we have the following expansion as x →∞
Setting q = 3 for instance, we get
Generalized Liu’s Formula
For any x > 0, we have
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.
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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
Setting x = 1 in this identity yields the following analogue of Fontana-Mascheroni’s series
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
Analogue of Wallis’s Product Formula
The following analogue of Wallis’s formula was already found in Project 10.1
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
namely
Analogue of Euler’s Series Representation of γ
We have ( Σψ)′(1) = −1 − γ and
The Taylor series expansion of Σψ(x + 1) about x = 0 is
Integrating both sides of this equation on (0, 1), we obtain
Analogue of the Reflection Formula
For any \(x\in \mathbb {R}\setminus \mathbb {Z}\), we have
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
If s = 2, then
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])
We also have
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
Hence we have the values
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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
In particular, we have
Generalized Liu’s Formula
For any x > 0 we have
Eulerian and Weierstrassian Forms
For any x > 0, we have
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
Setting x = 1 in this identity yields the analogue of Fontana-Mascheroni series
Analogue of Wallis’s Product Formula
The analogue of Wallis’s formula is
This formula is actually obtained by combining Proposition 6.7 with the duplication formula for the Hurwitz zeta function
On the other hand we also have (see Paris [83])
Combining this formula with the analogue of Wallis’s formula, we derive the following identity
Taylor Series Expansion
We have
The Taylor series expansion of ζ 2(s, x + 1) about x = 0 is
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
and li(x) is the logarithmic integral function defined for all positive real numbers x ≠ 1 by the integral
Incidentally, when p = 1, this function reduces to the ordinary generating function for the sequence n↦G n. That is,
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
with p − 1 leading 0’s.
We also note that the function h p has the following integral representation
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
The conversion formulas between the \(h_p^{\prime }s\) and the \(g_p^{\prime }s\) are simply given by the following equations
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)
that is,
with
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
see, e.g., Srivastava and Choi [93, p. 266]. Extending this definition to noninteger orders by writing
we obtain the following very compact integral representation
\(\lozenge \)
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
Using the change of variable u = 2s+p, we finally obtain
Now, we have
and hence the analogue of Raabe’s formula is
Generalized Stirling’s and Related Formulas When p = 1
For any a ≥ 0, we have the following limits and asymptotic equivalence as x →∞,
Upon differentiation,
where
Limit and Series Representations When p = 1
The Eulerian and Weierstrassian forms are
and
where
Gregory’s Formula-Based Series Representation When p = 1
Proposition 8.11 provides the following series representation: for any x > 0 we have
Setting x = 1 in this identity, we obtain the following analogue of Fontana-Mascheroni’s series
Analogue of Gauss’ Multiplication Formula
For any \(m\in \mathbb {N}^*\) and any x > 0, we have
Using the multiplication formula for the Hurwitz zeta function, we then obtain the following analogue of Gauss’ multiplication formula
Now, using (8.15) we obtain
Corollary 8.33 then tells us that the sequences
and
converge to the integrals
respectively.
References
V. Kowalenko. Properties and applications of the reciprocal logarithm numbers. Acta Appl. Math., 109(2):413–437, 2010.
R. B. Paris. A note on some infinite sums of Hurwitz zeta functions. Working note (arXiv:2104.00957v2), 2021.
H. M. Srivastava and J. Choi. Zeta and q-zeta functions and associated series and integrals. Elsevier, Inc., Amsterdam, 2012.
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Marichal, JL., Zenaïdi, N. (2022). Defining New Multiple \(\log \Gamma \)-Type Functions. In: A Generalization of Bohr-Mollerup's Theorem for Higher Order Convex Functions. Developments in Mathematics, vol 70. Springer, Cham. https://doi.org/10.1007/978-3-030-95088-0_11
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