Asymptotics of some generalized Mathieu series

TL;DR

This paper derives asymptotic estimates for generalized Mathieu series with power-logarithmic and factorial sequences, using Mellin transforms and Dirichlet series analysis, providing precise first-order asymptotics.

Contribution

It introduces a novel approach to asymptotic analysis of Mathieu series with complex sequences via Mellin transform techniques.

Findings

Power-logarithmic sequences yield precise first-order asymptotics.

Factorial sequences are analyzed despite natural boundary challenges.

Elementary estimates provide reasonably accurate asymptotics for factorial cases.

Abstract

We establish asymptotic estimates of Mathieu-type series defined by sequences with power-logarithmic or factorial behavior. By taking the Mellin transform, the problem is mapped to the singular behavior of certain Dirichlet series, which is then translated into asymptotics for the original series. In the case of power-logarithmic sequences, we obtain precise first order asymptotics. For factorial sequences, a natural boundary of the Mellin transform makes the problem more challenging, but a direct elementary estimate gives reasonably precise asymptotics.

Equations

$$S_{\mathbf{a},\mathbf{b},\mu}(r):=\sum_{n=0}^{\infty}\frac{a_{n}}{\big{(}b_{n}+r^{2}\big{)}^{\mu+1}}.$$

$$S_{\alpha,\beta,\gamma,\delta,\mu}(r):=\sum_{n=2}^{\infty}\frac{n^{\alpha}(\log n)^{\gamma}}{\big{(}n^{\beta}(\log n)^{\delta}+r^{2}\big{)}^{\mu+1}}.$$

$$S^{!}_{\alpha,\beta,\mu}(r):=\sum_{n=0}^{\infty}\frac{(n!)^{\alpha}}{\big{(}(n!)^{\beta}+r^{2}\big{)}^{\mu+1}},$$

$$\sum _{n=1}^{\infty }\frac{n}{(n^2+r^2{)}^2}\sim \sum _{k=0}^{\infty }(-1{)}^k\frac{B_{2k}}{2r^{2k+2}},r\uparrow \infty ,$$

$$\sum _{n=1}^{\infty }(-1{)}^{n-1}\frac{n}{(n^2+r^2{)}^2}\sim \sum _{k=1}^{\infty }\frac{G_{2k}}{4r^{2k+2}},r\uparrow \infty ,$$

$$\delta (\alpha +1)/\beta -\gamma \notin \mathbb{N}=\{1,2,\dots \},$$

$$C_{\alpha,\beta,\gamma,\delta,\mu}:=\frac{(\tfrac{1}{2}\beta)^{\delta(\alpha+1)/\beta-\gamma-1}\Gamma\big{(}\frac{\delta}{\beta}(\alpha+1)-\gamma+1\big{)}}{2\Gamma(\mu+1)\Gamma\big{(}{-\frac{\delta}{\beta}(\alpha+1)}+\gamma+1\big{)}}\\ \times\Gamma\big{(}{-\frac{\alpha+1}{\beta}}+\mu+1\big{)}\Gamma\big{(}\frac{\alpha+1}{\beta}\big{)}.$$

$$C_{\alpha,\beta,\gamma,\delta,\mu}:=\frac{\beta^{m-1}\Gamma\big{(}{-\frac{\alpha+1}{\beta}}+\mu+1\big{)}\Gamma\big{(}\frac{\alpha+1}{\beta}\big{)}}{2^{m}\Gamma(\mu+1)}.$$

$$S_{\alpha ,\beta ,\gamma ,\delta ,\mu }(r)\sim C_{\alpha ,\beta ,\gamma ,\delta ,\mu }{r}^{2(\alpha +1)/\beta -2(\mu +1)}(\log r{)}^{-\delta (\alpha +1)/\beta +\gamma },r\uparrow \infty .$$

$$2(\alpha+1)/\beta-2(\mu+1)=\frac{2}{\beta}\big{(}\alpha+1-\beta(\mu+1)\big{)}<0.$$

$$a_n\sim n^{\alpha }(\log n{)}^{\gamma },b_n\sim n^{\beta }(\log n{)}^{\delta },n\uparrow \infty .$$

$$S_{\mathbf{a},\mathbf{b},\mu }(r)\sim C_{\alpha ,\beta ,\gamma ,\delta ,\mu }{r}^{2(\alpha +1)/\beta -2(\mu +1)}(\log r{)}^{-\delta (\alpha +1)/\beta +\gamma },r\uparrow \infty .$$

$$S^{!}_{\alpha,\beta,\mu}(r)=r^{-2(\mu+1-\alpha/\beta)}\exp\big{(}{-m(r)}\log\log r+O(\log\log\log r)\big{)}$$

$$\mathcal{R}:=\big{\{}r>0:d_{1}\leq\{\Gamma^{-1}(r^{2/\beta})\}\leq d_{2}\big{\}},$$

$$m(r):=\min\Big{\{}\alpha\{\Gamma^{-1}(r^{2/\beta})\},\big{(}\beta(\mu+1)-\alpha\big{)}\big{(}1-\{\Gamma^{-1}(r^{2/\beta})\}\big{)}\Big{\}}>0.$$

$${\Gamma }^{-1}(r^{2/\beta })\sim \frac{2\log r}{\beta \log \log r},r\uparrow \infty .$$

$$r^{2\alpha /\beta -2(\mu +1)-\epsilon }\ll S_{\alpha ,\beta ,\mu }^{!}(r)\ll r^{2\alpha /\beta -2(\mu +1)+\epsilon },r\uparrow \infty ,$$

$$\log S_{\alpha ,\beta ,\mu }^{!}(r)=-2(\mu +1-\alpha /\beta )\log r+O(\log \log r),r\uparrow \infty .$$

$$S^{!}_{\alpha,\beta,\mu}(r)\leq r^{-2(\mu+1-\alpha/\beta)}\exp\big{(}o(\log\log r)\big{)},\quad r\to\infty.$$

$$S^{!}_{\alpha,\beta,\mu}(r)=O\Big{(}r^{-2(\mu+1-\alpha/\beta)}\frac{\log r}{\log\log r}\Big{)},\quad r\uparrow\infty.$$

$$S_{\mathbf{a},\mathbf{b},\mu}(r)=S_{\alpha,\beta,\gamma,\delta,\mu}(r)\big{(}1+o(1)\big{)}+O\big{(}r^{-2(\mu+1)}(\log r)^{2\alpha+1}\big{)},\quad r\uparrow\infty.$$

$$b_n+r^2$$

$$\displaystyle=r^{2}(1+O\big{(}(\log r)^{2\beta}/r^{2}\big{)}$$

$$\displaystyle=r^{2}\big{(}1+o(1)\big{)},\quad r\uparrow\infty,$$

$$(b_{n}+r^{2})^{-(\mu+1)}=r^{-2(\mu+1)}\big{(}1+o(1)\big{)},\quad 0\leq n\leq\lfloor\log r\rfloor.$$

$$\displaystyle\sum_{n=0}^{\lfloor\log r\rfloor}\frac{a_{n}}{\big{(}b_{n}+r^{2}\big{)}^{\mu+1}}$$

$$\sim r^{-2(\mu +1)}\sum _{n=0}^{\lfloor \log r\rfloor }{n}^{2\alpha }$$

$$\displaystyle=O\big{(}r^{-2(\mu+1)}(\log r)^{2\alpha+1}\big{)}.$$

$$b_n+r^2$$

$$\displaystyle=(n^{\beta}(\log n)^{\delta}+r^{2})\Big{(}1+\frac{o(n^{\beta}(\log n)^{\delta})}{n^{\beta}(\log n)^{\delta}+r^{2}}\Big{)}$$

$$\displaystyle=(n^{\beta}(\log n)^{\delta}+r^{2})\big{(}1+o(1)\big{)}.$$

$$a_{n}=n^{\alpha}(\log n)^{\gamma}\big{(}1+o(1)\big{)},\quad r\uparrow\infty.$$

$$\displaystyle\sum_{n>\lfloor\log r\rfloor}\frac{a_{n}}{\big{(}b_{n}+r^{2}\big{)}^{\mu+1}}$$

Full text

1 Introduction and main results

Define, for $\mu\geq 0,$ $r>0$ and sequences $\mathbf{a}=(a_{n})_{n\geq 0},$ $\mathbf{b}=(b_{n})_{n\geq 0},$

[TABLE]

The parametrization (i.e., $r^{2}$ and not $r$, $\mu+1$ and not $\mu$) is along the lines of [20]. Assumptions on the sequences $\mathbf{a}$ and $\mathbf{b}$ will be specified below. The study of such series began with 19th century work of Mathieu on elasticity of solid bodies, and has produced a considerable amount of literature, much of which focuses on integral representations and inequalities. See, e.g., [20, 21, 22] for some recent results and many references. As a special case of (1.1), define, for $\alpha,\beta,r>0,$ $\mu\geq 0$ with $\alpha-\beta(\mu+1)<-1$ and $\gamma,\delta\in\mathbb{R},$

[TABLE]

Note that the summation in (1.2) starts at $2$ to make the summand always well-defined. The series (1.2) is closely related to a paper by Paris [17] (see also [25]), but the presence of logarithmic factors is new. Another special case of (1.1) is the series

[TABLE]

defined for $\alpha,\mu\geq 0$, $\beta,r>0$ with $\alpha-\beta(\mu+1)<0.$ We are not aware of any asymptotic estimates for (1.3) in the literature. See [22] for integral representations for some series of this kind. The subject of the present paper is the asymptotic behavior of the Mathieu-type series (1.2) and (1.3) for $r\uparrow\infty$. For the classical Mathieu series, the asymptotic expansion

[TABLE]

was found be Elbert [6], whereas Pogány et al. [18] showed the expansion

[TABLE]

for its alternating counterpart; the $B_{n}$ and $G_{n}$ are Bernoulli resp. Genocchi numbers. We refer to [17] for further references on asymptotics of Mathieu-type series, to which we add §19 and §20 of [11]. To formulate our results on (1.2), for

[TABLE]

we define the constant

[TABLE]

If, on the other hand, $m:=\delta(\alpha+1)/\beta-\gamma\in\mathbb{N}$ is a positive integer, then we define

[TABLE]

Theorem 1.1**.**

Let $\alpha,\beta>0,$ $\mu\geq 0,$ with $\alpha-\beta(\mu+1)<-1$, and $\gamma,\delta\in\mathbb{R}$. Then we have

[TABLE]

Of course, the exponent of $r$ is negative:

[TABLE]

Also, we note that for $\gamma=\delta=0$ (no logarithmic factors), condition (1.4) is always satisfied, and the asymptotic equivalence (1.6) agrees with a special case of Theorem 3 in [17]. A bit more generally than Theorem 1.1, we have:

Theorem 1.2**.**

Let the parameters $\alpha,\beta,\gamma,\delta,\mu$ be as in Theorem 1.1. Let $\mathbf{a}$ and $\mathbf{b}$ be positive sequences that satisfy

[TABLE]

Then $S_{\mathbf{a},\mathbf{b},\mu}(r)$ has the asymptotic behavior stated in Theorem 1.1, i.e.

[TABLE]

This result includes sequences of the form $(\log n!)^{\alpha}$, see Corollary 6.1. Also, it clearly implies that shifts such as $a_{n}=(n+a)^{\alpha}(\log(n+b))^{\beta}$ are not visible in the first order asymptotics. Theorems 1.1 and 1.2 are proved in Section 2. The series (1.3) is more difficult to analyze than (1.2) by Mellin transform (see Section 3), but it turns out that it is asymptotically dominated by only two summands. This yields the following result, which is proved in Section 4. It uses an expansion for the functional inverse of the gamma function which is stated, but not proved in [2]; see Section 4 for details. We therefore state the theorem conditional on this expansion. We write $\{x\}$ for the fractional part of a real number $x$.

Theorem 1.3**.**

Assume that the expansion of the inverse gamma function stated in equation (70) of [2] is correct. Let $\alpha,\beta>0,$ $\mu\geq 0,$ with $\alpha-\beta(\mu+1)<0$, and $0<d_{1}<d_{2}<1$. Then

[TABLE]

as $r\to\infty$ in the set

[TABLE]

where the function $m(\cdot)$ is defined by

[TABLE]

Thus, under the constraint (1.8), the series $S^{!}_{\alpha,\beta,\mu}(r)$ decays like $r^{-2(\mu+1-\alpha/\beta)}$, accompanied by a power of $\log r$, where the exponent of the latter depends on $r$ and fluctuates in a finite interval of negative numbers. The expression inside the fractional part $\{\cdot\}$ growths roughly logarithmically:

[TABLE]

Clearly, the proportion $\lim_{r\uparrow\infty}r^{-1}\mathrm{meas}(\mathcal{R}\cap[0,r])$ of “good” values of $r$ can be made arbitrarily close to $1$ by choosing $d_{1}$ and $1-d_{2}$ sufficiently small. Without the Diophantine assumption (1.8), a more complicated asymptotic expression for $S^{!}_{\alpha,\beta,\mu}(r)$ is obtained by combining (4.2), (4.11), and (4.13) below. From this expression it is easy to see that, for any $\varepsilon>0$, we have

[TABLE]

as well as logarithmic asymptotics:

[TABLE]

The following result contains an asymptotic upper bound; like (1.10) and (1.11), it is valid without restricting $r$ to (1.8):

Theorem 1.4**.**

Assume that the expansion of the inverse gamma function stated in equation (70) of [2] is correct. Let $\alpha,\beta>0,$ $\mu\geq 0,$ with $\alpha-\beta(\mu+1)<0.$ Then

[TABLE]

Theorem 1.4 is proved in Section 4, too. In Section 3, we show the following unconditional bound, which also holds for $\alpha=0$:

Theorem 1.5**.**

Let $\alpha,\mu\geq 0,$ $\beta>0$ with $\alpha-\beta(\mu+1)<0.$ Then

[TABLE]

The difficulties concerning the factorial Mathieu-type series stem from the fact that the Mellin transform of $S^{!}_{\alpha,\beta,\mu}(\cdot)$ has a natural boundary in the form of a vertical line, whereas that of $S_{\alpha,\beta,\gamma,\delta,\mu}(\cdot)$ is more regular, featuring an analytic continuation with a single branch cut. See Sections 2 and 3 for details. We therefore prove Theorem 1.3 by a direct estimate; see Section 4. It will be clear from the proof that the error term in (1.7) can be refined, if desired. Also, $d_{1}$ and $1-d_{2}$ may depend on $r$, as long as they tend to zero sufficiently slowly.

2 Power-logarithmic sequences

Since (1.2) is a series with positive terms, the discrete Laplace method seems to be a natural asymptotic tool; see [16] for a good introduction and further references. However, while the summands of (1.2) do have a peak around $n\approx r^{2/\beta}$, the local expansion of the summand does not fully capture the asymptotics, and the central part of the sum yields an incorrect constant factor. A similar phenomenon has been observed in [5, 12] for integrals that are not amenable to the Laplace method. As in [17], we instead use a Mellin transform approach. Since the Mellin transform seems not to be explicitly available in our case, we invoke results from [13] on the analytic continuation of a certain Dirichlet series. Before beginning with the Mellin transform analysis, we show that Theorem 1.2 follows from Theorem 1.1. This is the content of the following lemma.

Lemma 2.1**.**

Let $\mathbf{a}$ and $\mathbf{b}$ be as in Theorem 1.2. Then

[TABLE]

Proof.

First consider the summation range $0\leq n\leq\lfloor\log r\rfloor$ for the series defining $S_{\mathbf{a},\mathbf{b},\mu}(r)$. We have the estimate

[TABLE]

and thus

[TABLE]

We obtain

[TABLE]

Now consider the range $\lfloor\log r\rfloor<n<\infty$, which yields the main contribution. As for the denominator, we have

[TABLE]

Note that the first two $o(\cdot)$ are meant for $n\uparrow\infty$, but then the term $\frac{o(n^{\beta}(\log n)^{\delta})}{n^{\beta}(\log n)^{\delta}+r^{2}}$ is also uniformly $o(1)$ as $r\uparrow\infty$, because $r\uparrow\infty$ implies $n\uparrow\infty$ in the range $\lfloor\log r\rfloor<n<\infty$. Similarly, we have

[TABLE]

Therefore,

[TABLE]

Here, the asymptotic equivalence follows from (2.2) and (2.3), and the equality follows from (2.1). The statement now follows by combining (2.1) and (2.4). ∎

We now begin the proof of Theorem 1.1. As in [13], define the Dirichlet series

[TABLE]

with real parameters $\eta,\theta$. We will see below that the Mellin transform of (1.2) can be expressed using $\zeta_{\eta,\theta}(s)$. The first two statements of the following lemma are taken from [13].

Lemma 2.2**.**

The Dirichlet series $\zeta_{\eta,\theta}$ has an analytic continuation to the whole complex plane except $(-\infty,1]$. As $s\to 1$ in this domain, we have the asymptotics

[TABLE]

The analytic continuation grows at most polynomially as $|\mathrm{Im}(s)|\uparrow\infty$ while $\mathrm{Re}(s)$ is bounded and positive.

Proof.

The statements about analytic continuation and asymptotics are proved in [13]. We revisit this proof in order to prove the polynomial estimate, which is needed later to apply Mellin inversion. By the Euler–Maclaurin summation formula, we have

[TABLE]

where

[TABLE]

and the $B$s are Bernoulli numbers resp. polynomials. As noted in [13], the last integral in (2.6) is holomorphic in $\mathrm{Re}(s)>-1$, and applying the Euler–Maclaurin formula of arbitrary order yields the full analytic continuation, after analyzing the first integral in (2.6). To prove our lemma, it remains to estimate the growth of the terms in (2.6). The dominating factor of $f^{(2)}(x)$ satisfies

[TABLE]

from which it is very easy to see that the last integral in (2.6) grows at most polynomially under the stated conditions on $s$. In the first integral in (2.6), we substitute

[TABLE]

(as in [13]) and obtain

[TABLE]

From Stirling’s formula, we have

[TABLE]

uniformly w.r.t. $\mathrm{Re}(t)$, as long as $\mathrm{Re}(t)$ stays bounded. Using this and

[TABLE]

we see that $|(s-1)^{\theta s-\eta-1}\Gamma(\eta-\theta s+1)|$ can be estimated by a polynomial in $s$. Finally, we have

[TABLE]

from which it is immediate that the term

[TABLE]

in (2.8) admits a polynomial estimate. ∎

For any sufficiently regular function $f$, we denote the Mellin transform by $f^{*}$,

[TABLE]

We now compute the Mellin transform of the function $S_{\alpha,\beta,\gamma,\delta,\mu}(r)$, writing $a_{n}=n^{\alpha}(\log n)^{\gamma}$ and $b_{n}=n^{\beta}(\log n)^{\delta}.$

[TABLE]

where we substituted $u=r^{2}/b_{n},$ and

[TABLE]

The Dirichlet series $D$ can be expressed in terms of $\zeta_{\eta,\theta}$ from (2.5):

[TABLE]

with

[TABLE]

Formula (2.9) is valid for $\mathrm{Re}(s)\in(0,\hat{s})$. The function $\Gamma(\mu+1-s/2)$ has poles at $2\mu+2,2\mu+4,\dots$, and those of $\Gamma(s/2)$ are $0,-2,-4,\dots$ All those poles are outside the strip $\{s\in\mathbb{C}:\mathrm{Re}(s)\in(0,\hat{s})\}.$ The singular expansion of (2.9) at the dominating singularity $\hat{s}$ can be translated, via the Mellin inversion formula, into the asymptotic behavior of $S_{\alpha,\beta,\gamma,\delta,\mu}(r)$. See [8] for a standard introduction to this method; in fact, our generalized Mathieu series (1.1) is a harmonic sum in the terminology of [8]. By Mellin inversion, we have

[TABLE]

where $\kappa\in(0,\hat{s})$. Note that integrability of $S^{*}_{\alpha,\beta,\gamma,\delta,\mu}(s)$ follows from the polynomial estimate in Lemma 2.2 and Stirling’s formula, as the latter implies

[TABLE]

for bounded $\mathrm{Re}(t)$. Suppose first that

[TABLE]

Then, from Lemma 2.2 and (2.10), we have

[TABLE]

with

[TABLE]

Combining (2.9) and (2.14) yields

[TABLE]

where

[TABLE]

By a standard procedure, we can now extract asymptotics of the Mathieu-type series $S_{\alpha,\beta,\gamma,\delta,\mu}(r)$ from (2.12). The integration contour in (2.12) is pushed to the right, which is allowed by Lemma 2.2. The real part of the new contour is

[TABLE]

where the singularity at $s=\hat{s}$ is avoided by a small C-shaped notch. In (2.18) below, this notch is the integration contour. The contour is then transformed to a Hankel contour $\mathcal{H}$ by the substitution $s=\hat{s}-w/\log r$. The contour $\mathcal{H}$ starts at $-\infty$, circles the origin counterclockwise and continues back to $-\infty$. Using (2.16), we thus obtain

[TABLE]

See [8, 9, 12, 13] for details of this asymptotic transfer. This completes the proof of (1.6) in the case $\delta(\alpha+1)/\beta-\gamma\notin\mathbb{N}$. Recall the definitions of the constants $\hat{s},c_{2},c_{3}$ in (2.11), (2.15), and (2.17).

Now suppose that

[TABLE]

We need to show that (1.6) still holds, but with the constant factor now given by (1.5). By Lemma 2.2 and (2.10), we have

[TABLE]

where

[TABLE]

Define

[TABLE]

Then, using (2.9) and (2.20),

[TABLE]

We proceed similarly as above (see again [8, 9, 13]) and find

[TABLE]

As for the second $\sim$, note that

[TABLE]

From the well-known residues of $\Gamma$ and $\psi$ at the non-positive integers (see, e.g., p.241 in [24]), we obtain

[TABLE]

Formula (1.6) is established, and Theorem 1.1 is proved. As for the constants in (2.22), recall the definitions in (2.11), (2.19), and (2.21). As mentioned above, Theorem 1.2 follows from Theorem 1.1 and Lemma 2.1.

3 Factorial sequences: the associated Dirichlet series

In the Mellin transform of (1.3), the following Dirichlet series occurs:

[TABLE]

As we will see in Lemma 3.1, this function does not have an analytic continuation beyond the right half-plane. It is well known that the presence of a natural boundary is a severe obstacle when doing asymptotic transfers; see [7] and the references cited there. Therefore, our proof of Theorem 1.3 in Section 4 will not use Mellin transform asymptotics. Still, some analytic properties of (3.1) seem to be interesting in their own right, and will be discussed in the present section. We note that the arguments at the beginning of the proof of Lemma 3.1 (analyticity, natural boundary) suffice to identify the location of the singularity of the Mellin transform of $S^{!}_{\alpha,\beta,\mu}(\cdot)$ (see (3.6) below), and thus yield the logarithmic asymptotics in (1.11) with the weaker error term $o(\log r)$. Moreover, in this section we will prove Theorem 1.5; see (3.15) below.

Lemma 3.1**.**

The function $\eta$ is analytic in the right half-plane, and the imaginary axis $i\mathbb{R}$ is a natural boundary. At the origin, we have the asymptotics

[TABLE]

Proof.

Analyticity follows from a standard result on Dirichlet series, see e.g. p.5 in [14]. As $n/\!\log n!=o(1)$, the lacunary series $\sum_{n=0}^{\infty}z^{\log n!}$ has the unit circle as a natural boundary. We refer to the introduction of [4] for details. This implies that $i\mathbb{R}$ is the natural boundary of

[TABLE]

It remains to prove (3.2). We begin by showing that the Dirichlet series

[TABLE]

has an analytic continuation to $\mathrm{Re}(s)>0$, with branch cut $(0,1]$. The main idea is that replacing $\log n!$ by $n\log n$ leads to the series from Lemma 2.2, and the properties of (3.3) that we need are the same as those stated there. We just do not care about continuation further left than $\mathrm{Re}(s)>0$, because we do not require it. The continuation of (3.3) is based on writing

[TABLE]

By Stirling’s formula, we have

[TABLE]

locally uniformly w.r.t. $s$ in the right half-plane. From this it follows that

[TABLE]

defines an analytic function of $s$ for $\mathrm{Re}(s)>0$. Moreover, the last series in (3.4) has an analytic continuation to a slit plane. This is proved by the same argument as in Lemma 2.2, using the Euler-Maclaurin formula and (2.7). Moreover, the polynomial estimate from that lemma easily extends to the continuation of (3.3) for $\mathrm{Re}(s)>0,$ $s\notin(0,1]$. After these preparations we can prove (3.2) by Mellin transform asymptotics. We compute, recalling the definition of $\zeta_{\eta,\theta}$ in (2.5) and its asymptotics from Lemma 2.2,

[TABLE]

We have shown above that the Dirichlet series in (3.5) has an analytic continuation to $\mathrm{Re}(t)>-1$, $t\notin(-1,0]$, and so Lemma 2 in [13] is applicable (asymptotic transfer, with $a=0$, $b=1$ in the notation of [13]). We conclude

[TABLE]

and hence

[TABLE]

Analogously to (2.9), we find the Mellin transform of (1.3):

[TABLE]

where

[TABLE]

By the Mellin inversion formula, we have

[TABLE]

Note that integrability of the Mellin transform $S^{!\,*}_{\alpha,\beta,\mu}$ follows from (2.13) and the obvious estimate

[TABLE]

By (3.6) and Lemma 3.1, the integrand in (3.8) has a singularity at $s=\tilde{s}$, with singular expansion

[TABLE]

It is well known that this kind of singularity (polynomial growth of the transform) is not amenable to the saddle point method, as regards precise asymptotics. Still, a saddle point bound can be readily found. For an introduction to saddle point bounds and the saddle point method, we recommend Chapter VIII in [10]. Retaining only the first two terms on the right-hand side of (3.10) and taking the derivative w.r.t. $s$ yields the saddle point equation

[TABLE]

with solution

[TABLE]

We take this as real part of the integration path in (3.8) and obtain, using (3.9),

[TABLE]

The fact that the integral is $O(1)$ as $r\uparrow\infty$ follows from (2.13). From (3.11), we have

[TABLE]

Lemma 3.1 implies

[TABLE]

which results in the saddle point bound

[TABLE]

which proves Theorem 1.5. Note that this bound is weaker than (1.12), but does not require the – so far not proven – expansion (4.9) of the inverse gamma function. The saddle point bound (3.15) also holds for $\alpha=0$, which is excluded in Theorems 1.3 and 1.4, because our proof of (4.4) below requires $\alpha>0$.

4 Factorial sequences: Proofs

This section contains the proofs of Theorems 1.3 and 1.4. Our estimates can be viewed as a somewhat degenerate instance of the Laplace method, where the central part of the sum consists of just two summands. We denote by $A_{n}$ the summands of (1.3):

[TABLE]

Define $n_{0}=n_{0}(r)$ by $n_{0}(r):=\lfloor\Gamma^{-1}(r^{2/\beta})\rfloor-1$, i.e.,

[TABLE]

We first show that $S^{!}_{\alpha,\beta,\mu}(r)$ is dominated by $A_{n_{0}}$ and $A_{n_{0}+1}$. For brevity, we omit writing the dependence of $A_{n}$ and $n_{0}$ on $r$.

Lemma 4.1**.**

Let $\alpha,\beta>0,$ $\mu\geq 0,$ with $\alpha-\beta(\mu+1)<0$. Then

[TABLE]

Proof.

For $k\geq 2$, we estimate, using (4.1),

[TABLE]

Therefore,

[TABLE]

This shows that

[TABLE]

For the initial segment $\sum_{k=1}^{n_{0}-1}A_{n_{0}-k}$ of the series, we use the following estimate for $k\geq 1$:

[TABLE]

Pick an integer $q$ with $q>1/\alpha$. Then

[TABLE]

Now $\sum_{k=1}^{q}B_{k}$ has a fixed number of summands, all $o(1)$, and is thus $o(1)$ as $r\uparrow\infty$. In the second sum, we pull out the factor $n_{0}^{-\alpha}$, estimate $q$ of the remaining factors by $n_{0}-k+1$, and the other factors by $1$:

[TABLE]

The last equality follows from $q>1/\alpha$. We conclude that (4.3) is $o(1)$, and thus

[TABLE]

which finishes the proof. ∎

We now evaluate $A_{n_{0}}$ and $A_{n_{0}+1}$ asymptotically. We use the following notation, partially in line with p.417f. of [2], where asymptotic inversion of the gamma function is discussed. We write $W(\cdot)$ for the Lambert $W$ function, which satisfies $W(z)\exp(W(z))=z$.

[TABLE]

It is easy to check, using the defining property of Lambert $W$, that

[TABLE]

in fact, this is equation (63) in [2].

Proof of Theorem 1.3.

By Stirling’s formula and (4.5), we have

[TABLE]

As mentioned in Theorems 1.3 and 1.4, we require the expansion

[TABLE]

of the inverse gamma function; see equation (70) in [2] (stated there, with an additional term, but without proof). Note that first order asymptotics $\Gamma^{-1}(x)\sim u_{0}$, i.e. (1.9), are very easy to prove using the approach of [2], just by carrying the $O(1/u)$ term neglected after equation (62) in [2] a few lines further. From (4.8) and (4.9), we obtain

[TABLE]

Together with (4.7), this yields

[TABLE]

Equation (4.10) is crucial for determining the asymptotics of the right hand side of (4.2). Since

[TABLE]

we can use (4.10) to evaluate the summand $A_{n_{0}}$ as

[TABLE]

In the last line, we used the fact that

[TABLE]

see [3]. The definition of $n_{0}$ (see (4.1)), (4.9), and (4.12) imply

[TABLE]

As for the summand $A_{n_{0}+1}$, we thus have (with $\log^{3}=\log\log\log$)

[TABLE]

This holds as $r\to\infty$, without any constraints on $r$. If $\{g\}\leq d_{2}<1$, as assumed in Theorem 1.3, then the term inside the big parentheses in (4.13) goes to infinity; note that $\log u_{0}\sim\log\log r$ by (4.6) and (4.12). We then have

[TABLE]

Define

[TABLE]

and

[TABLE]

Then, by Lemma 4.1, (4.11), and (4.14), we obtain

[TABLE]

Theorem 1.3 now follows from this, (4.11), and (4.14). Note that the assumption $0<d_{1}\leq\{g\}\leq d_{2}<1$ of Theorem 1.3 ensures that the term $(\dots)\log\log r$ in (4.11) and (4.14) asymptotically dominates the error term. Moreover, the asymptotic equivalence in (4.15) and (4.16) can be replaced by an equality, because the error factor $1+o(1)$ is absorbed into the $O(\log^{3}r)$ in the exponent. ∎

Proof of Theorem 1.4.

By (4.11), we have

[TABLE]

and so, by Lemma 4.1, it suffices to estimate $A_{n_{0}+1}$. Fix an arbitrary $\varepsilon>0$. Recall the notation introduced around (4.5). If $r$ is such that $\alpha(1-\{g\})\leq\varepsilon$, then we simply estimate the term in big parentheses in (4.13) by $1$, and obtain

[TABLE]

If, on the other hand, $\{g\}<1-\varepsilon/\alpha$, then (4.14) holds, which implies

[TABLE]

because the quantity in front of $\log\log r$ in (4.14) is negative. We have thus shown that, for any $\varepsilon>0$,

[TABLE]

From this, Theorem 1.4 easily follows. Indeed, were it not true, then there would be $\varepsilon^{\prime}>0$ and a sequence $r_{n}\uparrow\infty$ such that

[TABLE]

contradicting (4.17). ∎

5 Power sequences: Full expansion in a special case

In [20], an integral representation of the generalized Mathieu series

[TABLE]

was derived. In our notation, this series is

[TABLE]

We use said integral representation and Watson’s lemma to find a full expansion of $S_{\mu}(r)$ as $r\to\infty$. This expansion is not new (see Theorem 1 in [17]), and so we do not give full details. Still, our approach provides an independent check for (a special case of) Theorem 1 in [17], and it might be useful for other Mathieu-type series admitting a representation as a Laplace transform. The integral representation in Theorem 4 of [20] is

[TABLE]

where

[TABLE]

and $g_{\mu}$ is the Schlömilch series

[TABLE]

For $\mathrm{Re}(s)>\tfrac{3}{2}-\mu$, the Mellin transform of $g_{\mu}$ is

[TABLE]

The factor $\zeta(s+\mu-1/2)$ has a pole at $\check{s}:=\tfrac{3}{2}-\mu$, and $\Gamma(\mu/2+1/4+s/2)$ has poles at $s_{k}:=-2k-\mu-\tfrac{1}{2}$, $k\in\mathbb{N}_{0}$. By using Mellin inversion and collecting residues, we find that the expansion of $g_{\mu}(t)$ as $t\downarrow 0$ is

[TABLE]

No we multiply this expansion by $t^{\mu+1/2}$ and use Watson’s lemma ([15], p.71) in (5.1). In the notation of [15], p.71, the parameters $\mu$ and $\lambda$ are $\tfrac{1}{2}$ and our $\mu$, respectively. Simplifying the resulting expansion using Legendre’s duplication formula,

[TABLE]

yields the expansion

[TABLE]

as $r\to\infty$. Recall that the values of the zeta function at negative odd integers can be represented by Bernoulli numbers:

[TABLE]

The expansion (5.2) indeed agrees with Theorem 1 in [17], and the first term agrees with our Theorem 1.1 (with $\alpha=1$, $\beta=2$, $\gamma=\delta=0$). The divergent series in (5.2) looks very similar to formula (3.2) in [19], but there the argument of $\zeta(\cdot)$ in the summation is eventually positive instead of negative.

Finally, we give an amusing non-rigorous derivation of the asymptotic series on the right-hand side of (5.2), by using the binomial theorem, the “formula” $\zeta(-2k-1)=\sum_{n=1}^{\infty}n^{2k+1},$ and interchanging summation:

[TABLE]

Note that the dominating term of order $r^{-2\mu}$ is not found by this heuristic.

6 Application and further comments

We now apply Theorem 1.2 (on power-logarithmic sequences) to an example taken from [22]. There, integral representations for some Mathieu-type series were deduced, and we state asymptotics for one of them.

Corollary 6.1**.**

Let $\alpha,\beta>0,$ $\mu\geq 0,$ with $\alpha-\beta(\mu+1)<-1$. Then

[TABLE]

with

[TABLE]

Proof.

By Stirling’s formula, we have $(\log n!)^{\alpha}\sim(n\log n)^{\alpha}$. The statement thus follows from Theorem 1.2, with $\gamma=\alpha,$ $\delta=\beta,$ and $m=\delta(\alpha+1)/\beta-\gamma=1\in\mathbb{N}.$ ∎

A natural generalization of our main results on power-logarithmic sequences (Theorems 1.1 and 1.2) would be to replace $\log$ by an arbitrary slowly varying function: $a_{n}=n^{\alpha}\ell_{1}(n)$, $b_{n}=n^{\beta}\ell_{2}(n)$. Then the Dirichlet series (2.10) becomes

[TABLE]

The dominating singularity is still $\hat{s}$ defined in (2.11), as follows from Proposition 1.3.6 in [1], but it seems not easy to determine the singular behavior of $D$ at $\hat{s}$ for generic $\ell_{1},\ell_{2}$. Still, for specific examples such as $(\log\log n)^{\gamma}$ or $\exp(\sqrt{\log n})$, this should be doable. Note that our second step, i.e. the asymptotic transfer from the Mellin transform to the original function, works for slowly varying functions under mild conditions; see [9].

Finally, we note that introducing a geometrically decaying factor $x^{n}$ to the series (1.1) leads to a Mathieu-type power series. According to the following proposition, its asymptotics can be found in an elementary way, for rather general sequences $\mathbf{a},\mathbf{b}$. We refer to [23] for integral representations and further references on certain Mathieu-type power series.

Proposition 6.2**.**

Let $x\in\mathbb{C}$ with $|x|<1$, $a_{n}\in\mathbb{C}$, $b_{n}\geq 0,$ and $\mu\geq 0$. If $\sum_{n=0}^{\infty}a_{n}x^{n}$ is absolutely convergent and $b_{n}\uparrow\infty$, then

[TABLE]

Proof.

We have

[TABLE]

As $\sum_{n:\,b_{n}>r}|a_{n}||x|^{n}$ tends to zero, this is $o(r^{-2(\mu+1)})$. For the dominating part of the series, we find

[TABLE]

In the last equality, we used that $\sum_{n:\,b_{n}>r}a_{n}x^{n}=o(1)$, because $b_{n}\uparrow\infty$. ∎

In Proposition 6.2, we assumed $|x|<1$. Our main results (Theorems 1.1–1.3) are concerned with the case $x=1$, for some special sequences $\mathbf{a},\mathbf{b}$. An alternating factor $(-1)^{n}$, on the other hand, induces cancellations that are difficult to handle, and usually requires the availability of an explicit Mellin transform, as in [17].