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.
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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])
Equivalently, we have G 0(x) = x and
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
Just as for the gamma function and the Barnes G-function, we can derive the following asymptotic equivalence: for any a ≥ 0,
with equality if a ∈{0, 1, …, p}. We also have the following product representation
and the recurrence formula
For example, one can show that
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
For any s > 0, we define the function \(g_s\colon \mathbb {R}_+\to \mathbb {R}\) by
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
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
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
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
The generalized Stirling formula yields the following limit
Incidentally, the analogue of Legendre’s duplication formula provides the surprising identity
12.4 The Exponential Integral
Recall that the exponential integral E 1(x) is defined by the equation
Similarly to the previous example, let us consider the function \(g\colon \mathbb {R}_+\to \mathbb {R}\) defined by the equation
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
The generalized Stirling formula easily provides the following convergence result
Moreover, the analogue of Raabe’s formula is
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
we immediately derive (see also Example 8.21)
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
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
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
It follows that the modified function
satisfies the difference equation
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
where B k is the kth Bernoulli number. This polynomial satisfies the difference equation
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
that is, in view of (10.16)
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
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:
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
In particular, we have ψ n(0) = G n. Also, these polynomials satisfy the difference equation
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
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 \(c_n\in \mathbb {R}\).
References
N. J. A. Sloane (editor). The on-line encyclopedia of integer sequences. http://www.oeis.org
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). Further Examples. 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_12
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