1 Introduction

In 1933, Bendersky [4] studied the product \(\prod _{k=1}^nk^{k^q}\) for \(q=0,1,2,\ldots \), which reduces to the classical factorial function n! when \(q=0\) and the classical hyperfactorial function \(H(n)=\prod _{k=1}^nk^k\) when \(q=1\). He examined the logarithm of the product and determined the first five values of the limits

$$\begin{aligned} \ln (A_q)=\lim _{n\rightarrow \infty }\ln (A_q(n)) =\lim _{n\rightarrow \infty }\left\{ \sum _{k=1}^nk^q\ln k-P_q(n)\right\} , \end{aligned}$$
(1.1)

where

$$\begin{aligned} P_q(n)&=\frac{n^q}{2}\ln n+\frac{n^{q+1}}{q+1}\left( \ln n-\frac{1}{q+1}\right) \\&\quad +q!\sum _{j=1}^q\frac{n^{q-j}B_{j+1}}{(j+1)!(q-j)!} \left\{ \ln n+(1-\delta _{q,j})\sum _{l=1}^j\frac{1}{q-l+1}\right\} , \end{aligned}$$

\(B_n\) are the Bernoulli numbers, and \(\delta _{q,j}\) is the Kronecker delta function defined by \(\delta _{q,j}=0\) for \(j\ne q\) and \(\delta _{q,j}=1\) for \(j=q\). In 1995 and 1998, Choudhury [16] and Adamchik [2] showed independently that the constants \(A_q\) can be expressed in terms of the derivatives of the Riemann zeta function \(\zeta (s)\)

$$\begin{aligned} A_q=\exp \left\{ \frac{B_{q+1}H_q}{q+1}-\zeta '(-q)\right\} , \end{aligned}$$

where \(H_n\) are the harmonic numbers.

From (1.1), it follows that

$$\begin{aligned} \ln A_0&=\lim _{n\rightarrow \infty }\ln (A_0(n)) =\lim _{n\rightarrow \infty }\left\{ \sum _{k=1}^n\ln k-\left( n+\frac{1}{2}\right) \ln n+n\right\} ,\\ \ln A_1&=\lim _{n\rightarrow \infty }\ln (A_1(n)) =\lim _{n\rightarrow \infty }\left\{ \sum _{k=1}^nk\ln k -\left( \frac{n^2}{2}+\frac{n}{2}+\frac{1}{12}\right) \ln n+\frac{n^2}{4}\right\} , \end{aligned}$$

which indicate that \(A_0=\sqrt{2\pi }\) and \(A_1\) is the Glaisher–Kinkelin constant. The Glaisher–Kinkelin constant \(A_1=1.2824271291\ldots \) is closely related to the Barnes G-function G(z) by the limit

$$\begin{aligned} A_1=\lim _{n\rightarrow \infty } \frac{(2\pi )^{\frac{n}{2}}n^{\frac{n^2}{2}-\frac{1}{12}} \mathrm {e}^{-\frac{3n^2}{4}+\frac{1}{12}}}{G(n+1)}, \end{aligned}$$

and satisfies many beautiful formulas; see Finch’s book [20, Sect. 2.15].

Moreover, for \(q=2,3\), Eq. (1.1) gives

$$\begin{aligned} \ln A_2&=\lim _{n\rightarrow \infty }\ln (A_2(n)) \\&=\lim _{n\rightarrow \infty }\left\{ \sum _{k=1}^nk^2\ln k -\left( \frac{n^3}{3}+\frac{n^2}{2}+\frac{n}{6}\right) \ln n+\frac{n^3}{9}-\frac{n}{12}\right\} ,\\ \ln A_3&=\lim _{n\rightarrow \infty }\ln (A_3(n)) \\&=\lim _{n\rightarrow \infty }\left\{ \sum _{k=1}^nk^3\ln k -\left( \frac{n^4}{4}+\frac{n^3}{2}+\frac{n^2}{4}-\frac{1}{120}\right) \ln n +\frac{n^4}{16}-\frac{n^2}{12}\right\} , \end{aligned}$$

where \(A_2=1.0309167521\ldots \) and \(A_3=0.9795555269\ldots \). According to Finch’s book [20, Sect. 2.15] and the On-Line Encyclopedia of Integer Sequences (OEIS), the constants \(A_q\) should be called the generalized Glaisher–Kinkelin constants or the Bendersky constants.

The generalized Glaisher–Kinkelin constants have been used in the closed-form evaluation of some series involving zeta functions and in calculation of some integrals of multiple gamma functions; see Choi and Srivastava’s works [12, 14, 15, 36]. Recently, many researches are devoted to establishing asymptotic expansions on these constants and the related hyperfactorial functions, and the readers are referred to the papers [5, 7, 10, 11, 13, 22, 23, 26, 29, 38].

In particular, Chen [5] presented in 2012 the asymptotic expansions of \(\ln A_1(n)\), \(\ln A_2(n)\), and \(\ln A_3(n)\) by using the Euler–Maclaurin summation formula. For example, the expansion of \(\ln A_1(n)\) is

$$\begin{aligned} \ln A_1(n)\sim \ln A_1 -\sum _{k=1}^{\infty }\frac{B_{2k+2}}{2k(2k+1)(2k+2)}\frac{1}{n^{2k}},\quad n\rightarrow \infty . \end{aligned}$$

Substituting the values of \(B_n\) and using the expression of \(\ln A_1(n)\), the above expansion can be written as

$$\begin{aligned} \prod _{k=1}^nk^k&\sim A_1\cdot n^{\frac{n^2}{2}+\frac{n}{2}+\frac{1}{12}}\mathrm {e}^{-\frac{n^2}{4}} \exp \left( \frac{1}{720n^2}-\frac{1}{5040n^4}+\frac{1}{10080n^6}\right. \nonumber \\&\quad \left. -\frac{1}{9504n^8}+\frac{691}{3603600n^{10}}-\frac{1}{1872n^{12}}+\cdots \right) ,\quad n\rightarrow \infty . \end{aligned}$$
(1.2)

Mortici [26] established (1.2) and gave a recurrence relation to compute the coefficients of the series in the formula. Using (1.2), Chen and Lin [10] obtained a general asymptotic expansion

$$\begin{aligned} \prod _{k=1}^nk^k\sim A_1\cdot n^{\frac{n^2}{2}+\frac{n}{2}+\frac{1}{12}}\mathrm {e}^{-\frac{n^2}{4}} \left( \sum _{k=0}^{\infty }\frac{\check{\alpha }_k}{n^k}\right) ^{\frac{1}{r}},\quad n\rightarrow \infty , \end{aligned}$$
(1.3)

and presented the expression of \((\check{\alpha }_k)\). Wang and Liu [38] gave two general expansions

$$\begin{aligned}&\prod _{k=1}^nk^k\sim A_1\cdot n^{\frac{n^2}{2}+\frac{n}{2}+\frac{1}{12}}\mathrm {e}^{-\frac{n^2}{4}} \left( \sum _{k=0}^{\infty }\frac{\alpha _k}{(n+h)^k}\right) ^{\frac{1}{r}}, \end{aligned}$$
(1.4)
$$\begin{aligned}&\prod _{k=1}^nk^k\sim A_1\cdot n^{\frac{n^2}{2}+\frac{n}{2}+\frac{1}{12}}\mathrm {e}^{-\frac{n^2}{4}} \left( \sum _{k=0}^{\infty }\frac{\varphi _k}{(n+h)^k}\right) ^{\frac{n}{r}+q}, \end{aligned}$$
(1.5)

as \(n\rightarrow \infty \), and studied systematically the recurrences and the explicit expressions of \((\alpha _k)\) and \((\varphi _k)\). Moreover, Choi [13] presented the expression

$$\begin{aligned} \ln A_q(n)&=\sum _{k=1}^nk^q\ln k \nonumber \\&\quad -\left\{ \frac{n^{q+1}}{q+1}+\frac{n^q}{2} +\sum _{r=1}^{[\frac{q+1}{2}]}\frac{B_{2r}}{(2r)!} \left( \prod _{j=1}^{2r-1}(q-j+1)\right) n^{q+1-2r}\right\} \ln n\nonumber \\&\quad +\frac{n^{q+1}}{(q+1)^2} \nonumber \\&\quad -\sum _{r=1}^{[\frac{q+1}{2}]+\frac{(-1)^q-1}{2}} \frac{B_{2r}}{(2r)!}\left\{ \prod _{j=1}^{2r-1}(q-j+1)\sum _{j=1}^{2r-1}\frac{1}{q-j+1}\right\} n^{q+1-2r}\nonumber \\&=\sum _{k=1}^nk^q\ln k-U_{q+1}(n)\ln n+V_{q+1}(n), \end{aligned}$$
(1.6)

and gave the general asymptotic expansion

$$\begin{aligned} \ln A_q(n)\sim \ln A_q+(-1)^qq!\sum _{r=[\frac{q+1}{2}]+1}^{\infty } \frac{B_{2r}}{(2r)!}\cdot \frac{(2r-q-2)!}{n^{2r-q-1}},\quad n\rightarrow \infty , \end{aligned}$$
(1.7)

which reduces to Stirling’s formula of n! when \(q=0\) and Chen’s results in [5] when \(q=1,2,3\). Further results may be found in Chen [7], Cheng and Chen [11], Lin [22], Lu and Mortici [23], and Mortici [29].

Inspired by these works, we present in this paper the next two general asymptotic expansions on the hyperfactorial functions and the generalized Glaisher–Kinkelin constants

$$\begin{aligned}&\prod _{k=1}^nk^{k^q}\sim A_q\cdot n^{U_{q+1}(n)}\mathrm {e}^{-V_{q+1}(n)} \left( \sum _{k=0}^{\infty }\frac{\alpha _k(q;h,r)}{(n+h)^k}\right) ^{\frac{1}{r}},\end{aligned}$$
(1.8)
$$\begin{aligned}&\prod _{k=1}^nk^{k^q}\sim A_q\cdot n^{U_{q+1}(n)}\mathrm {e}^{-V_{q+1}(n)} \left( \sum _{k=0}^{\infty }\frac{\varphi _k(q;h,r,s)}{(n+h)^k}\right) ^{\frac{n}{r}+s}, \end{aligned}$$
(1.9)

as \(n\rightarrow \infty \), where the polynomials \(U_{q+1}(n)\) and \(V_{q+1}(n)\) are defined in (1.6). We give recurrences and explicit expressions of the coefficient sequences in the expansions by the exponential complete Bell polynomials, and discuss some special cases of these two expansions.

In particular, when \(q=0\), our results reduce to the asymptotic expansions of n!, including as special cases some well-known formulas due to Laplace, Wehmeier, and Ramanujan, and some recent results due to Batir [3], Chen [6, 8, 9], Mortici [28, 30], Nemes [31, 32], et al. Additionally, when \(q=1\), our results reduce to the asymptotic expansions on the classical hyperfactorial function \(\prod _{k=1}^nk^k\) and the classical Glaisher–Kinkelin constant \(A_1\), including those presented by Chen and Lin [10] and Wang and Liu [38].

The paper is organized as follows: Sections 2 and 3 are devoted to the first general asymptotic expansion (1.8), and Sects. 4 and 5 are devoted to the second one (1.9).

2 The first general asymptotic expansion

Define the exponential complete Bell polynomials \(Y_n\) by

$$\begin{aligned} \exp \left( \sum _{m=1}^{\infty }x_m\frac{t^m}{m!}\right) =\sum _{n=0}^{\infty }Y_n(x_1,x_2,\ldots ,x_n)\frac{t^n}{n!}\,; \end{aligned}$$
(2.1)

see [18, Sect. 3.3] and [34, Sect. 2.8]. Then \(Y_0=1\) and

$$\begin{aligned} Y_n(x_1,x_2,\ldots ,x_n)=\sum _{c_1+2c_2+\cdots +nc_n=n} \frac{n!}{c_1!c_2!\cdots c_n!} \left( \frac{x_1}{1!}\right) ^{c_1}\left( \frac{x_2}{2!}\right) ^{c_2}\cdots \left( \frac{x_n}{n!}\right) ^{c_n}. \end{aligned}$$
(2.2)

According to [34, Eq. (2.44)] (see also [17, Eq. (3.6)] and [35, Theorem 1]), the polynomials \(Y_n\) satisfy the recurrence

$$\begin{aligned} Y_n(x_1,x_2,\ldots ,x_n)=\sum _{j=0}^{n-1}\left( {\begin{array}{c}n-1\\ j\end{array}}\right) x_{n-j}Y_j(x_1,x_2,\ldots ,x_j) ,\quad n\ge 1. \end{aligned}$$
(2.3)

Using the definition and recurrence of the Bell polynomials, the following general asymptotic expansion can be obtained.

Theorem 2.1

Let h, r be real numbers such that \(r\ne 0\). Define the sequence \((\beta _k)_{k\ge 1}\) by

$$\begin{aligned} \beta _k=(-1)^qr(k-1)!\left( {\begin{array}{c}k+q\\ q\end{array}}\right) ^{-1}\frac{B_{k+q+1}}{k+q+1}. \end{aligned}$$
(2.4)

Then

$$\begin{aligned} \prod _{k=1}^nk^{k^q}\sim A_q\cdot n^{U_{q+1}(n)}\mathrm {e}^{-V_{q+1}(n)} \left( \sum _{k=0}^{\infty }\frac{\alpha _k(q;h,r)}{(n+h)^k}\right) ^{\frac{1}{r}}, \end{aligned}$$
(2.5)

as \(n\rightarrow \infty \), where \((\alpha _k(q;h,r))_{k\ge 0}\) is determined by

$$\begin{aligned} \alpha _0(q;h,r)=1,\quad \alpha _k(q;h,r)=\frac{y_k}{k!}-\sum _{j=0}^{k-1}(-1)^{k-j}\left( {\begin{array}{c}k-1\\ k-j\end{array}}\right) h^{k-j}\alpha _j(q;h,r),\quad k\ge 1, \end{aligned}$$
(2.6)

and \((y_k)_{k\ge 0}\) is determined by

$$\begin{aligned} y_0=1,\quad y_k=\sum _{j=0}^{k-1}\left( {\begin{array}{c}k-1\\ j\end{array}}\right) \beta _{k-j}y_j,\quad k\ge 1. \end{aligned}$$
(2.7)

Proof

Define the falling factorials \((x)_n\) by \((x)_0=1\) and \((x)_n=x(x-1)\cdots (x-n+1)\) for \(n=1,2,\ldots \). The asymptotic expansion (1.7) can be rewritten as

$$\begin{aligned} \prod _{k=1}^nk^{k^q}\sim A_q\cdot n^{U_{q+1}(n)}\mathrm {e}^{-V_{q+1}(n)} \exp \left\{ \sum _{k=1}^{\infty }\frac{(-1)^qq!B_{k+q+1}}{(k+q+1)_{q+2}}\frac{1}{n^k}\right\} , \quad n\rightarrow \infty .\quad \end{aligned}$$
(2.8)

Then (2.1) and (2.4) give

$$\begin{aligned} \left( \frac{\prod _{k=1}^nk^{k^q}}{A_q\cdot n^{U_{q+1}(n)}\mathrm {e}^{-V_{q+1}(n)}}\right) ^r&\sim \exp \left\{ \sum _{k=1}^{\infty }\frac{(-1)^qrk!q!B_{k+q+1}}{(k+q+1)_{q+2}} \frac{(\frac{1}{n})^k}{k!}\right\} \\&=\exp \left\{ \sum _{k=1}^{\infty }\beta _k\frac{(\frac{1}{n})^k}{k!}\right\} \\ \nonumber&=\sum _{k=0}^{\infty }\frac{Y_k(\beta _1,\beta _2,\ldots ,\beta _k)}{k!}\frac{1}{n^k}, \quad n\rightarrow \infty . \end{aligned}$$

On the other hand, expansion in powers of 1 / n yields

$$\begin{aligned} \sum _{j=0}^{\infty }\frac{\alpha _j(q;h,r)}{(n+h)^j}&=\sum _{j=0}^{\infty }\frac{\alpha _j(q;h,r)}{n^j(1+h/n)^j} =\sum _{j=0}^{\infty }\frac{\alpha _j(q;h,r)}{n^j} \sum _{i=0}^{\infty }\left( {\begin{array}{c}-j\\ i\end{array}}\right) \frac{h^i}{n^i}\\&=\sum _{k=0}^{\infty }\left\{ \sum _{j=0}^k(-1)^{k-j} \left( {\begin{array}{c}k-1\\ k-j\end{array}}\right) h^{k-j}\alpha _j(q;h,r)\right\} \frac{1}{n^k}. \end{aligned}$$

Thus, it suffices to show that the system

$$\begin{aligned} \frac{Y_k(\beta _1,\beta _2,\ldots ,\beta _k)}{k!} =\sum _{j=0}^k(-1)^{k-j}\left( {\begin{array}{c}k-1\\ k-j\end{array}}\right) h^{k-j}\alpha _j(q;h,r) \end{aligned}$$

has unique solution \((\alpha _k(q;h,r))_{k\ge 0}\). This is established next. The case \(k=0\) gives \(\alpha _0(q;h,r)=1\). For \(k\ge 1\), the system gives

$$\begin{aligned} \alpha _k(q;h,r)=\frac{Y_k(\beta _1,\beta _2,\ldots ,\beta _k)}{k!} -\sum _{j=0}^{k-1}(-1)^{k-j}\left( {\begin{array}{c}k-1\\ k-j\end{array}}\right) h^{k-j}\alpha _j(q;h,r). \end{aligned}$$

Setting \(y_k=Y_k(\beta _1,\beta _2,\ldots ,\beta _k)\) gives recurrence (2.6) and shows that \((\alpha _k(q;h,r))\) can be uniquely determined. Finally, (2.3) gives (2.7) and the proof is complete. \(\square \)

By specifying the parameters q, h, r in Theorem 2.1, many special asymptotic expansions on \(\prod _{k=1}^nk^{k^q}\) and \(A_q\) can be obtained. In particular, when \(h=0\), Theorem 2.1 reduces to the following result.

Theorem 2.2

Let \(r\ne 0\) be a real number. Then

$$\begin{aligned} \prod _{k=1}^nk^{k^q}\sim A_q\cdot n^{U_{q+1}(n)}\mathrm {e}^{-V_{q+1}(n)} \left( \sum _{k=0}^{\infty }\frac{\alpha _k(q;0,r)}{n^k}\right) ^{\frac{1}{r}}, \end{aligned}$$
(2.9)

as \(n\rightarrow \infty \), where \((\alpha _k(q;0,r))_{k\ge 0}\) is determined by

$$\begin{aligned}&\alpha _0(q;0,r)=1,\\&\alpha _k(q;0,r)=\frac{(-1)^qr}{k}\sum _{j=0}^{k-1}\left( {\begin{array}{c}k-j+q\\ q\end{array}}\right) ^{-1} \frac{B_{k-j+q+1}}{k-j+q+1}\alpha _j(q;0,r),\quad k\ge 1. \end{aligned}$$

The further special cases \(q=0\) and \(q=2\) are stated next.

Corollary 2.3

Let \(r\ne 0\) be a real number. Then

$$\begin{aligned} n!\sim \sqrt{2\pi n}\left( \frac{n}{\mathrm {e}}\right) ^n \left( \sum _{k=0}^{\infty }\frac{\alpha _k(0;0,r)}{n^k}\right) ^{\frac{1}{r}}, \end{aligned}$$
(2.10)

as \(n\rightarrow \infty \), where \((\alpha _k(0;0,r))_{k\ge 0}\) is determined by

$$\begin{aligned} \alpha _0(0;0,r)=1,\quad \alpha _k(0;0,r)=\frac{r}{k}\sum _{j=0}^{k-1}\frac{B_{k-j+1}}{k-j+1}\alpha _j(0;0,r), \quad k\ge 1. \end{aligned}$$
(2.11)

Corollary 2.4

Let \(r\ne 0\) be a real number. Then

$$\begin{aligned} \prod _{k=1}^nk^{k^2}\sim A_2\cdot n^{\frac{n^3}{3}+\frac{n^2}{2}+\frac{n}{6}}\mathrm {e}^{-\frac{n^3}{9}+\frac{n}{12}} \left( \sum _{k=0}^{\infty }\frac{\alpha _k(2;0,r)}{n^k}\right) ^{\frac{1}{r}}, \end{aligned}$$
(2.12)

as \(n\rightarrow \infty \), where \((\alpha _k(2;0,r))_{k\ge 0}\) is determined by

$$\begin{aligned}&\alpha _0(2;0,r)=1,\nonumber \\&\alpha _k(2;0,r)=\frac{2r}{k}\sum _{j=0}^{k-1}\frac{B_{k-j+3}}{(k-j+1)(k-j+2)(k-j+3)} \alpha _j(2;0,r),\quad k\ge 1. \end{aligned}$$
(2.13)

The recurrences (2.11) and (2.13) determine the coefficients \((\alpha _k(0;0,r))\) and \((\alpha _k(2;0,r))\), respectively. For example,

$$\begin{aligned}&\alpha _0(0;0,r)=1,\quad \alpha _1(0;0,r)=\frac{r}{12},\quad \alpha _2(0;0,r)=\frac{r^2}{288},\\&\alpha _3(0;0,r)=\frac{r(-144+5r^2)}{51840},\quad \alpha _4(0;0,r)=\frac{r^2(-576+5r^2)}{2488320}, \end{aligned}$$

and

$$\begin{aligned}&\alpha _0(2;0,r)=1,\quad \alpha _1(2;0,r)=-\frac{r}{360},\quad \alpha _2(2;0,r)=\frac{r^2}{259200},\\&\alpha _3(2;0,r)=-\frac{r(-259200+7r^2)}{1959552000},\quad \alpha _4(2;0,r)=\frac{r^2(-1036800+7r^2)}{2821754880000}. \end{aligned}$$

Example 2.1

Setting \(r=1\) in Corollary 2.3 yields

$$\begin{aligned} n!\sim \sqrt{2\pi n}\left( \frac{n}{\mathrm {e}}\right) ^n \left( 1+\frac{1}{12n}+\frac{1}{288n^2} -\frac{139}{51840n^3} -\frac{571}{2488320n^4} +\frac{163879}{209018880n^5}+\cdots \right) , \end{aligned}$$

as \(n\rightarrow \infty \), which is the famous Laplace formula, and sometimes called Stirling’s formula (see [19, pp. 2–3] ). Setting \(r=2\) in Corollary 2.3 gives the Wehmeier formula

$$\begin{aligned} n!\sim \sqrt{2\pi n}\left( \frac{n}{\mathrm {e}}\right) ^n \left( 1+\frac{1}{6n}+\frac{1}{72n^2}-\frac{31}{6480n^3}-\frac{139}{155520n^4} +\frac{9871}{6531840n^5}+\cdots \right) ^{\frac{1}{2}}, \end{aligned}$$

as \(n\rightarrow \infty \), which was recently rediscovered by Batir [3], Luschny [25], and Mortici [27]. Setting \(r=6\) gives the well-known Ramanujan formula

$$\begin{aligned} n!\sim \sqrt{2\pi n}\left( \frac{n}{\mathrm {e}}\right) ^n \left( 1+\frac{1}{2n}+\frac{1}{8n^2}+\frac{1}{240n^3}-\frac{11}{1920n^4} +\frac{79}{26880n^5}+\cdots \right) ^{\frac{1}{6}}, \end{aligned}$$

as \(n\rightarrow \infty \) (see, for example, [21, 33]). Batir [3] obtained the case \(r=4\), Mortici [28, 30] presented the cases \(r=8,10,12,14\), and Chen and Lin [6, 9] gave the cases \(r=\frac{1}{2},\frac{1}{4},\frac{1}{6}\), and \(r=-1,-2\).

In 2013, Lu and Wang [24] studied the expansion (2.10), but they determined only the first five terms of the coefficient sequence \((\alpha _k(0;0,r))\) and did not obtain an explicit expression nor recurrence relation for it. Chen and Lin [6, 9] also presented (2.10) and established the expression for \((\alpha _k(0;0,r))\). In 2016, Wang further gave the recurrence (2.11) for \((\alpha _k(0;0,r))\), showed a more general expansion, and generalized Lu, Wang, Chen and Lin’s results (see [37, Theorem 2.1 and Corollary 3.3]).

Example 2.2

Setting \(r=1\) in Corollary 2.4 yields

$$\begin{aligned} \prod _{k=1}^nk^{k^2}&\sim A_2\cdot n^{\frac{n^3}{3}+\frac{n^2}{2}+\frac{n}{6}}\mathrm {e}^{-\frac{n^3}{9}+\frac{n}{12}} \left( 1-\frac{1}{360n}+\frac{1}{259200n^2}+\frac{259193}{1959552000n^3}\right. \\&\quad \left. -\frac{1036793}{2821754880000n^4}-\frac{201551328007}{5079158784000000n^5} +\cdots \right) ,\quad n\rightarrow \infty . \end{aligned}$$

Other special cases of Theorems 2.1 and 2.2 and Corollaries 2.3 and 2.4 can be obtained similarly.

3 Explicit expression of the coefficient sequence in (2.5)

In this section, the method of generating functions is used to present an explicit expression for the coefficients in the asymptotic expansion (2.5).

Theorem 3.1

The coefficient sequence \((\alpha _k(q;h,r))\) in (2.5) is given by the Bell polynomials

$$\begin{aligned} \alpha _k(q;h,r)=\frac{1}{k!}Y_k(\delta _1,\delta _2,\ldots ,\delta _k), \end{aligned}$$

where

$$\begin{aligned} \delta _k=\frac{(-1)^qrk!q!}{(k+q+1)_{q+2}} \left\{ B_{k+q+1}(h)-\sum _{j=0}^{q+1}\left( {\begin{array}{c}k+q+1\\ j\end{array}}\right) B_jh^{k+q+1-j}\right\} ,\quad k\ge 1, \end{aligned}$$

and \(B_n(x)\) are the classical Bernoulli polynomials.

Proof

Introduce the notations \((\beta _k)_{k\ge 1}\), \((y_k)_{k\ge 0}\) and \((\alpha _k(q;h,r))_{k\ge 0}\) by

$$\begin{aligned} f_{\beta }(t)=\sum _{k=1}^{\infty }\beta _k\frac{t^k}{k!},\quad f_y(t)=\sum _{k=0}^{\infty }y_k\frac{t^k}{k!},\quad f_{\alpha }(t)=\sum _{k=0}^{\infty }\alpha _k(q;h,r)t^k. \end{aligned}$$

The result \(f_y'(t)=f_y(t)\cdot f_{\beta }'(t)\) comes from (2.7). Then \(f_y(t)=C\cdot \mathrm {e}^{f_{\beta }(t)}\). The initial conditions \(f_y(0)=1\) and \(f_{\beta }(0)=0\) show that \(C=1\). Therefore \(f_y(t)=\exp (f_{\beta }(t))\). On the other hand, (2.6) gives

$$\begin{aligned} f_y(t)&=\sum _{k=0}^{\infty }\frac{y_k}{k!}t^k =\sum _{k=0}^{\infty }\sum _{j=0}^k(-1)^{k-j}\left( {\begin{array}{c}k-1\\ k-j\end{array}}\right) h^{k-j}\alpha _j(q;h,r)t^k\\&=\sum _{j=0}^{\infty }\alpha _j(q;h,r)t^j\sum _{i=0}^{\infty }\left( {\begin{array}{c}-j\\ i\end{array}}\right) (ht)^i\\ \nonumber&=\sum _{j=0}^{\infty }\alpha _j(q;h,r)\left( \frac{t}{1+ht}\right) ^j=f_{\alpha }\left( \frac{t}{1+ht}\right) . \end{aligned}$$

Then

$$\begin{aligned} f_{\alpha }(t)=f_y\left( \frac{t}{1-ht}\right) =\exp \left\{ f_{\beta }\left( \frac{t}{1-ht}\right) \right\} . \end{aligned}$$
(3.1)

From the definition of \((\beta _k)\) and the identity for Bernoulli polynomials

$$\begin{aligned} B_n(x)=\sum _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) B_kx^{n-k} \end{aligned}$$

(see [1, Chap. 23] and [18, Sect. 1.14]), it follows that

$$\begin{aligned}&f_{\beta }\left( \frac{t}{1-ht}\right) =\sum _{k=1}^{\infty }(-1)^qr(k-1)!\left( {\begin{array}{c}k+q\\ q\end{array}}\right) ^{-1}\frac{B_{k+q+1}}{k+q+1} \frac{(\frac{t}{1-ht})^k}{k!}\\&\quad =(-1)^qrq!\sum _{k=1}^{\infty }\frac{(k-1)!B_{k+q+1}t^k}{(k+q+1)!} \sum _{j=0}^{\infty }\left( {\begin{array}{c}-k\\ j\end{array}}\right) (-ht)^j\\&\quad =(-1)^qrq!\sum _{m=1}^{\infty }\sum _{k=1}^m \frac{(k-1)!B_{k+q+1}}{(k+q+1)!}\left( {\begin{array}{c}m-1\\ m-k\end{array}}\right) h^{m-k}t^m\\&\quad =(-1)^qrq!\sum _{m=1}^{\infty }\left\{ \frac{1}{(m+q+1)_{q+2}} \sum _{k=1}^m\left( {\begin{array}{c}m+q+1\\ k+q+1\end{array}}\right) B_{k+q+1}h^{m-k}\right\} t^m\\&\quad =(-1)^qrq!\sum _{m=1}^{\infty }\frac{t^m}{(m+q+1)_{q+2}}\\ \nonumber&\qquad \times \left\{ B_{m+q+1}(h)-\sum _{j=0}^{q+1}\left( {\begin{array}{c}m+q+1\\ j\end{array}}\right) B_jh^{m+q+1-j}\right\} . \end{aligned}$$

Define the coefficient of \(t^m/m!\) in the last series by \(\delta _m\). Then

$$\begin{aligned} f_{\alpha }(t)=\exp \left( \sum _{m=1}^{\infty }\delta _m\frac{t^m}{m!}\right) , \end{aligned}$$

and the expression of \((\alpha _k)\) follows from here. \(\square \)

The special case \(h=0\) gives

$$\begin{aligned} \tilde{\delta }_m=\frac{(-1)^qrm!q!}{(m+q+1)_{q+2}}B_{m+q+1}. \end{aligned}$$

This produces the next corollary.

Corollary 3.2

The coefficient sequence \((\alpha _k(q;0,r))\) in (2.9) has the explicit expression

$$\begin{aligned} \alpha _k(q;0,r)&=\sum _{c_1+2c_2+\cdots +kc_k=k} \frac{((-1)^qrq!)^{c_1+c_2+\cdots +c_k}}{c_1!c_2!\cdots c_k!}\\ \nonumber&\quad \times \left( \frac{B_{q+2}}{(q+2)_{q+2}}\right) ^{c_1} \left( \frac{B_{q+3}}{(q+3)_{q+2}}\right) ^{c_2} \cdots \\&\quad \times \left( \frac{B_{q+k+1}}{(q+k+1)_{q+2}}\right) ^{c_k}. \end{aligned}$$

Moreover, when \(h=0\) and q is odd, using \(B_{2k+1}=0\) for \(k=1,2,\ldots \) gives

$$\begin{aligned} f_{\tilde{\alpha }}(t)&=\sum _{k=0}^{\infty }\alpha _k(q;0,r)t^k =\exp \left( \sum _{m=1}^{\infty }\tilde{\delta }_m\frac{t^m}{m!}\right) \\ \nonumber&=\exp \left\{ \sum _{m=1}^{\infty }\frac{-rq!}{(m+q+1)_{q+2}}B_{m+q+1}t^m\right\} \\&=\exp \left\{ \sum _{k=1}^{\infty }\frac{-rq!}{(2k+q+1)_{q+2}}B_{2k+q+1}t^{2k}\right\} . \end{aligned}$$

This shows that in this case \(f_{\tilde{\alpha }}(t)\) is an even function and \(\alpha _{2k+1}(q;0,r)=0\) for \(k=0,1,2,\ldots \). Thus, Theorem 2.2 and Corollary 3.2 produce the following result.

Theorem 3.3

Let q be an odd integer and \(r\ne 0\) be a real number. Then

$$\begin{aligned} \prod _{k=1}^nk^{k^q}\sim A_q\cdot n^{U_{q+1}(n)}\mathrm {e}^{-V_{q+1}(n)} \left( \sum _{k=0}^{\infty }\frac{\gamma _k(q;0,r)}{n^{2k}}\right) ^{\frac{1}{r}},\quad n\rightarrow \infty . \end{aligned}$$

The sequence \((\gamma _k(q;0,r))_{k\ge 0}\) satisfies the recurrence

$$\begin{aligned}&\gamma _0(q;0,r)=1,\\&\gamma _k(q;0,r)=-\frac{r}{2k}\sum _{j=0}^{k-1}\left( {\begin{array}{c}2k-2j+q\\ q\end{array}}\right) ^{-1} \frac{B_{2k-2j+q+1}}{2k-2j+q+1}\gamma _j(q;0,r),\quad k\ge 1, \end{aligned}$$

and has the explicit expression

$$\begin{aligned} \gamma _k(q;0,r)&=\sum _{d_1+2d_2+\cdots +kd_k=k} \frac{(-rq!)^{d_1+d_2+\cdots +d_k}}{d_1!d_2!\cdots d_k!}\\ \nonumber&\quad \times \left( \frac{B_{q+3}}{(q+3)_{q+2}}\right) ^{d_1} \left( \frac{B_{q+5}}{(q+5)_{q+2}}\right) ^{d_2} \cdots \left( \frac{B_{q+2k+1}}{(q+2k+1)_{q+2}}\right) ^{d_k}. \end{aligned}$$

Proof

Set \(\gamma _k(q;0,r)=\alpha _{2k}(q;0,r)\). Then the theorem follows from the recurrence and expression of \(\alpha _{2k}(q;0,r)\) as well as the vanishing of \(B_{2k+1}\). \(\square \)

The special cases \(q=1\) and \(q=3\) in Theorem 3.3 are stated in Corollaries 3.4 and 3.5.

Corollary 3.4

Let \(r\ne 0\) be a real number. Then

$$\begin{aligned} \prod _{k=1}^nk^k\sim A_1\cdot n^{\frac{n^2}{2}+\frac{n}{2}+\frac{1}{12}} \mathrm {e}^{-\frac{n^2}{4}} \left( \sum _{k=0}^{\infty }\frac{\gamma _k(1;0,r)}{n^{2k}}\right) ^{\frac{1}{r}}, \end{aligned}$$
(3.2)

as \(n\rightarrow \infty \), where \((\gamma _k(1;0,r))_{k\ge 0}\) is determined by

$$\begin{aligned}&\gamma _0(1;0,r)=1,\nonumber \\&\gamma _k(1;0,r)=-\frac{r}{2k}\sum _{j=0}^{k-1}\frac{B_{2k-2j+2}}{(2k-2j+1)(2k-2j+2)} \gamma _j(1;0,r),\quad k\ge 1. \end{aligned}$$
(3.3)

Corollary 3.5

Let \(r\ne 0\) be a real number. Then

$$\begin{aligned} \prod _{k=1}^nk^{k^3} \sim A_3\cdot n^{\frac{n^4}{4}+\frac{n^3}{2}+\frac{n^2}{4}-\frac{1}{120}} \mathrm {e}^{-\frac{n^4}{16}+\frac{n^2}{12}} \left( \sum _{k=0}^{\infty }\frac{\gamma _k(3;0,r)}{n^{2k}}\right) ^{\frac{1}{r}}, \end{aligned}$$
(3.4)

as \(n\rightarrow \infty \), where \((\gamma _k(3;0,r))_{k\ge 0}\) is determined by

$$\begin{aligned} \gamma _0(3;0,r)=1,\quad \gamma _k(3;0,r)=-\frac{3r}{k}\sum _{j=0}^{k-1}\frac{B_{2k-2j+4}}{(2k-2j+4)_4} \gamma _j(3;0,r),\quad k\ge 1.\quad \end{aligned}$$
(3.5)

Using these recurrences, the coefficients \((\gamma _k(1;0,r))\) and \((\gamma _k(3;0,r))\) can be computed efficiently. For example,

$$\begin{aligned}&\gamma _0(1;0,r)=1,\quad \gamma _1(1;0,r)=\frac{r}{720},\quad \gamma _2(1;0,r)=\frac{r(-1440+7r)}{7257600},\\&\gamma _3(1;0,r)=\frac{r(1555200-4320r+7r^2)}{15676416000},\\&\gamma _4(1;0,r)=\frac{r(-365783040000+547430400r-665280r^2+539r^3)}{3476402012160000}, \end{aligned}$$

and

$$\begin{aligned}&\gamma _0(3;0,r)=1,\quad \gamma _1(3;0,r)=-\frac{r}{5040},\quad \gamma _2(3;0,r)=\frac{r(1512+r)}{50803200},\\&\gamma _3(3;0,r)=-\frac{r(127008000+49896r+11r^2)}{8449588224000},\\&\gamma _4(3;0,r)=\frac{r(35385851289600+7585171776r+1297296r^2+143r^3)}{2214468081745920000}. \end{aligned}$$

Example 3.1

Setting \(r=1\) in Corollaries 3.4 and 3.5 gives

$$\begin{aligned} \prod _{k=1}^nk^k&\sim A_1\cdot n^{\frac{n^2}{2}+\frac{n}{2}+\frac{1}{12}}\mathrm {e}^{-\frac{n^2}{4}} \left( 1+\frac{1}{720n^2}-\frac{1433}{7257600n^4}+\frac{1550887}{15676416000n^6} \right. \\&\quad \left. -\frac{365236274341}{3476402012160000n^8} +\frac{31170363588856607}{162695614169088000000n^{10}} -\cdots \right) \end{aligned}$$

and

$$\begin{aligned} \prod _{k=1}^nk^{k^3}&\sim A_3\cdot n^{\frac{n^4}{4}+\frac{n^3}{2}+\frac{n^2}{4}-\frac{1}{120}} \mathrm {e}^{-\frac{n^4}{16}+\frac{n^2}{12}}\\ \nonumber&\quad \times \left( 1-\frac{1}{5040n^2}+\frac{1513}{50803200n^4} -\frac{127057907}{8449588224000n^6}\right. \\&\quad \left. +\frac{7078687551763}{442893616349184000n^8} -\frac{1626209947417109183}{55804595659997184000000n^{10}}+\cdots \right) , \end{aligned}$$

as \(n\rightarrow \infty \). Other special cases can be obtained similarly. Corollary 3.4 and some of its special cases have been presented in [38, Corollary 2.2 and Example 2.1]. See also Chen and Lin [10].

4 The second general asymptotic expansion

This section presents another general asymptotic expansion for the hyperfactorial functions.

Theorem 4.1

Let h, r, s be real numbers such that \(r\ne 0\). Define the sequence \((\psi _m)_{m\ge 1}\) by

$$\begin{aligned} \psi _1=0,\quad \psi _m=m!\sum _{k=1}^{m-1}\frac{(-1)^{q+m-k-1}r^{m-k}s^{m-k-1}q!B_{k+q+1}}{(k+q+1)_{q+2}} ,\quad m\ge 2. \end{aligned}$$
(4.1)

Then

$$\begin{aligned} \prod _{k=1}^nk^{k^q}\sim A_q\cdot n^{U_{q+1}(n)}\mathrm {e}^{-V_{q+1}(n)} \left( \sum _{k=0}^{\infty }\frac{\varphi _k(q;h,r,s)}{(n+h)^k}\right) ^{\frac{n}{r}+s}, \end{aligned}$$
(4.2)

as \(n\rightarrow \infty \), where \(U_{q+1}(n)\) and \(V_{q+1}(n)\) are defined in (1.6), \((\varphi _k(q;h,r,s))_{k\ge 0}\) is determined by

$$\begin{aligned}&\varphi _0(q;h,r,s)=1,\nonumber \\&\varphi _k(q;h,r,s)=\frac{z_k}{k!}-\sum _{j=0}^{k-1}(-1)^{k-j} \left( {\begin{array}{c}k-1\\ k-j\end{array}}\right) h^{k-j}\varphi _j(q;h,r,s),\quad k\ge 1, \end{aligned}$$
(4.3)

and \((z_k)_{k\ge 0}\) is determined by

$$\begin{aligned} z_0=1,\quad z_k=\sum _{j=0}^{k-2}\left( {\begin{array}{c}k-1\\ j\end{array}}\right) \psi _{k-j}z_j,\quad k\ge 1. \end{aligned}$$
(4.4)

Proof

From (2.8) and the definition of \((\psi _m)\), it follows that

$$\begin{aligned}&\left( \frac{\prod _{k=1}^nk^{k^q}}{A_q\cdot n^{U_{q+1}(n)} \mathrm {e}^{-V_{q+1}(n)}}\right) ^{\frac{r}{n+rs}} \sim \exp \left\{ \frac{r}{n+rs} \sum _{k=1}^{\infty }\frac{(-1)^qq!B_{k+q+1}}{(k+q+1)_{q+2}}\frac{1}{n^k}\right\} \\&\quad =\exp \left\{ \frac{r}{n}\sum _{j=0}^{\infty }\left( -\frac{rs}{n}\right) ^j \sum _{k=1}^{\infty }\frac{(-1)^qq!B_{k+q+1}}{(k+q+1)_{q+2}}\frac{1}{n^k}\right\} \\&\quad =\exp \left\{ \sum _{m=2}^{\infty }\sum _{k=1}^{m-1} \frac{(-1)^{q+m-k-1}r^{m-k}s^{m-k-1}q!B_{k+q+1}}{(k+q+1)_{q+2}} \frac{1}{n^m}\right\} \\&\quad =\exp \left\{ \sum _{m=1}^{\infty }\psi _m\frac{\left( \frac{1}{n}\right) ^m}{m!}\right\} =\sum _{k=0}^{\infty }\frac{Y_k(\psi _1,\psi _2,\ldots ,\psi _k)}{k!}\frac{1}{n^k}, \quad n\rightarrow \infty . \end{aligned}$$

Moreover,

$$\begin{aligned} \sum _{k=0}^{\infty }\frac{\varphi _k(q;h,r,s)}{(n+h)^k} =\sum _{k=0}^{\infty }\left\{ \sum _{j=0}^k(-1)^{k-j} \left( {\begin{array}{c}k-1\\ k-j\end{array}}\right) h^{k-j}\varphi _j(q;h,r,s)\right\} \frac{1}{n^k}. \end{aligned}$$

Define \(z_k=Y_k(\psi _1,\psi _2,\ldots ,\psi _k)\). Using the same procedure as in the proof of Theorem 2.1, it can be verified that the system

$$\begin{aligned} \frac{z_k}{k!}=\sum _{j=0}^k(-1)^{k-j}\left( {\begin{array}{c}k-1\\ k-j\end{array}}\right) h^{k-j}\varphi _j(q;h,r,s) \end{aligned}$$

has unique solution \((\varphi _k(q;h,r,s))\). This can be computed by recurrences (4.3) and (4.4). \(\square \)

In the case \(h=0\) and \(s=0\), Theorem 4.1 gives

Theorem 4.2

Let \(r\ne 0\) be a real number. Then

$$\begin{aligned} \prod _{k=1}^nk^{k^q}\sim A_q\cdot n^{U_{q+1}(n)}\mathrm {e}^{-V_{q+1}(n)} \left( \sum _{k=0}^{\infty }\frac{\varphi _k(q;0,r,0)}{n^k}\right) ^{\frac{n}{r}}, \end{aligned}$$
(4.5)

as \(n\rightarrow \infty \), where \((\varphi _k(q;0,r,0))_{k\ge 0}\) is determined by

$$\begin{aligned}&\varphi _0(q;0,r,0)=1,\\&\varphi _k(q;0,r,0)=\frac{(-1)^qrq!}{k} \sum _{j=0}^{k-2}\frac{(k-j)B_{k-j+q}}{(k-j+q)_{q+2}}\varphi _j(q;0,r,0) ,\quad k\ge 1. \end{aligned}$$

Proof

In this case, \(\tilde{z}_k=k!\varphi _k(q;0,r,0)\) and

$$\begin{aligned} \tilde{\psi }_1=0,\quad \tilde{\psi }_m=\frac{(-1)^qrm!q!B_{m+q}}{(m+q)_{q+2}},\quad m\ge 2. \end{aligned}$$
(4.6)

By (4.4), the result follows. \(\square \)

Setting \(q=1\) and \(q=3\) in Theorem 4.2 yields the next two corollaries.

Corollary 4.3

Let \(r\ne 0\) be a real number. Then

$$\begin{aligned} \prod _{k=1}^nk^k \sim A_1\cdot n^{\frac{n^2}{2}+\frac{n}{2}+\frac{1}{12}}\mathrm {e}^{-\frac{n^2}{4}} \left( \sum _{k=0}^{\infty }\frac{\varphi _k(1;0,r,0)}{n^k}\right) ^{\frac{n}{r}}, \end{aligned}$$
(4.7)

as \(n\rightarrow \infty \), where \((\varphi _k(1;0,r,0))_{k\ge 0}\) is determined by

$$\begin{aligned}&\varphi _0(1;0,r,0)=1,\nonumber \\&\varphi _k(1;0,r,0)=-\frac{r}{k}\sum _{j=0}^{k-2}\frac{B_{k-j+1}}{(k-j+1)(k-j-1)} \varphi _j(1;0,r,0),\quad k\ge 1. \end{aligned}$$
(4.8)

Corollary 4.4

Let \(r\ne 0\) be a real number. Then

$$\begin{aligned} \prod _{k=1}^nk^{k^3} \sim A_3\cdot n^{\frac{n^4}{4}+\frac{n^3}{2}+\frac{n^2}{4}-\frac{1}{120}} \mathrm {e}^{-\frac{n^4}{16}+\frac{n^2}{12}} \left( \sum _{k=0}^{\infty }\frac{\varphi _k(3;0,r,0)}{n^k}\right) ^{\frac{n}{r}}, \end{aligned}$$
(4.9)

as \(n\rightarrow \infty \), where \((\varphi _k(3;0,r,0))_{k\ge 0}\) is determined by

$$\begin{aligned} \varphi _0(3;0,r,0)=1,\quad \varphi _k(3;0,r,0)=-\frac{6r}{k}\sum _{j=0}^{k-2}\frac{(k-j)B_{k-j+3}}{(k-j+3)_5} \varphi _j(3;0,r,0),\quad k\ge 1. \end{aligned}$$
(4.10)

The first few terms of \((\varphi _k(1;0,r,0))\) are

$$\begin{aligned}&\varphi _0(1;0,r,0)=1,\quad \varphi _1(1;0,r,0)=0,\quad \varphi _2(1;0,r,0)=0,\quad \varphi _3(1;0,r,0)=\frac{r}{720},\\&\varphi _4(1;0,r,0)=0,\quad \varphi _5(1;0,r,0)=-\frac{r}{5040},\quad \varphi _6(1;0,r,0)=\frac{r^2}{1036800}, \end{aligned}$$

and the first few terms of \((\varphi _k(3;0,r,0))\) are

$$\begin{aligned}&\varphi _0(3;0,r,0)=1,\quad \varphi _1(3;0,r,0)=0,\quad \varphi _2(3;0,r,0)=0,\quad \varphi _3(3;0,r,0)=-\frac{r}{5040},\\&\varphi _4(3;0,r,0)=0,\quad \varphi _5(3;0,r,0)=\frac{r}{33600},\quad \varphi _6(3;0,r,0)=\frac{r^2}{50803200}. \end{aligned}$$

Example 4.1

In the case \(r=1\), Corollaries 4.3 and 4.4 produce

$$\begin{aligned} \prod _{k=1}^nk^k&\sim A_1\cdot n^{\frac{n^2}{2}+\frac{n}{2}+\frac{1}{12}}\mathrm {e}^{-\frac{n^2}{4}} \left( 1+\frac{1}{720n^3}-\frac{1}{5040n^5}+\frac{1}{1036800n^6}\right. \\&\quad \left. +\frac{1}{10080n^7}-\frac{1}{3628800n^8}-\frac{2591989}{24634368000n^9} +\cdots \right) ^n \end{aligned}$$

and

$$\begin{aligned} \prod _{k=1}^nk^{k^3}&\sim A_3\cdot n^{\frac{n^4}{4}+\frac{n^3}{2}+\frac{n^2}{4}-\frac{1}{120}} \mathrm {e}^{-\frac{n^4}{16}+\frac{n^2}{12}}\left( 1-\frac{1}{5040n^3}+\frac{1}{33600n^5} +\frac{1}{50803200n^6}\right. \\&\quad \left. -\frac{1}{66528n^7}-\frac{1}{169344000n^8}+\frac{1755250417}{109844646912000n^9} +\cdots \right) ^n, \end{aligned}$$

as \(n\rightarrow \infty \). Corollary 4.3 and its special cases appear as [38, Corollary 4.2 and Example 4.1]. See also Chen [7, Remark 2].

5 Explicit expression of the coefficient sequence in (4.2)

Now define

$$\begin{aligned} f_{\psi }(t)=\sum _{k=1}^{\infty }\psi _k\frac{t^k}{k!},\quad f_z(t)=\sum _{k=0}^{\infty }z_k\frac{t^k}{k!},\quad f_{\varphi }(t)=\sum _{k=0}^{\infty }\varphi _k(q;h,r,s)t^k. \end{aligned}$$

As in the discussion above,

$$\begin{aligned}&f_{\varphi }(t)=f_z\left( \frac{t}{1-ht}\right) =\exp \left\{ f_{\psi }\left( \frac{t}{1-ht}\right) \right\} ,\\&f_{\psi }(t)=(-1)^qrq!\sum _{k=1}^{\infty }\frac{B_{k+q+1}}{(k+q+1)_{q+2}} \cdot \frac{t^{k+1}}{1+rst}, \end{aligned}$$

which give an explicit expression of \((\varphi _k(q;h,r,s))\) in terms of the Bell polynomials. In particular, when \(h=0\) and \(s=0\),

$$\begin{aligned} f_{\tilde{\varphi }}(t)=\sum _{k=0}^{\infty }\varphi _k(q;0,r,0)t^k =\exp \{f_{\tilde{\psi }}(t)\} =\exp \left( \sum _{m=1}^{\infty }\tilde{\psi }_m\frac{t^m}{m!}\right) , \end{aligned}$$

where the sequence \((\tilde{\psi }_m)\) is defined in (4.6). Then the following result holds.

Theorem 5.1

The coefficient sequence \((\varphi _k(q;0,r,0))\) in (4.5) is

$$\begin{aligned} \varphi _k(q;0,r,0)&=\frac{1}{k!}Y_k(\tilde{\psi }_1,\tilde{\psi }_2,\ldots ,\tilde{\psi }_k)\\&=\sum _{2c_2+3c_3+\cdots +kc_k=k} \frac{((-1)^qrq!)^{c_2+c_3+\cdots +c_k}}{c_2!c_3!\cdots c_k!}\\ \nonumber&\quad \times \left( \frac{B_{q+2}}{(q+2)_{q+2}}\right) ^{c_2} \left( \frac{B_{q+3}}{(q+3)_{q+2}}\right) ^{c_3} \cdots \left( \frac{B_{q+k}}{(q+k)_{q+2}}\right) ^{c_k}. \end{aligned}$$

Moreover, when \(h=0\), \(s=0\), and q is even, we have

$$\begin{aligned} f_{\tilde{\varphi }}(t)&=\sum _{k=0}^{\infty }\varphi _k(q;0,r,0)t^k =\exp \left\{ rq!\sum _{m=2}^{\infty }\frac{B_{m+q}}{(m+q)_{q+2}}t^m\right\} \\&=\exp \left\{ rq!\sum _{j=1}^{\infty }\frac{B_{2j+q}}{(2j+q)_{q+2}}t^{2j}\right\} . \end{aligned}$$

In this case \(f_{\tilde{\varphi }}(t)\) is an even function and \(\varphi _{2k+1}(q;0,r,0)=0\) for \(k=0,1,2,\ldots \). Now, defining \(\omega _k(q;0,r,0)=\varphi _{2k}(q;0,r,0)\), we obtain the next result.

Theorem 5.2

Let q be an even integer and \(r\ne 0\) be a real number. Then

$$\begin{aligned} \prod _{k=1}^nk^{k^q}\sim A_q\cdot n^{U_{q+1}(n)}\mathrm {e}^{-V_{q+1}(n)} \left( \sum _{k=0}^{\infty }\frac{\omega _k(q;0,r,0)}{n^{2k}}\right) ^{\frac{n}{r}},\quad n\rightarrow \infty . \end{aligned}$$

The sequence \((\omega _k(q;0,r,0))_{k\ge 0}\) satisfies the recurrence

$$\begin{aligned}&\omega _0(q;0,r,0)=1,\\&\omega _k(q;0,r,0)=\frac{rq!}{k} \sum _{j=0}^{k-1}\frac{(k-j)B_{2k-2j+q}}{(2k-2j+q)_{q+2}} \omega _{j}(q;0,r,0),\quad k\ge 1, \end{aligned}$$

and has the explicit expression

$$\begin{aligned} \omega _k(q;0,r,0)&=\sum _{d_1+2d_2+\cdots +kd_k=k} \frac{(rq!)^{d_1+d_2+\cdots +d_k}}{d_1!d_2!\cdots d_k!} \left( \frac{B_{q+2}}{(q+2)_{q+2}}\right) ^{d_1} \\&\quad \times \left( \frac{B_{q+4}}{(q+4)_{q+2}}\right) ^{d_2} \cdots \left( \frac{B_{q+2k}}{(q+2k)_{q+2}}\right) ^{d_k}. \end{aligned}$$

The special cases \(q=0\) and \(q=2\) in Theorem 5.2 are stated next.

Corollary 5.3

Let \(r\ne 0\) be a real number. Then

$$\begin{aligned} n!\sim \sqrt{2\pi n}\left( \frac{n}{\mathrm {e}}\right) ^n \left( \sum _{k=0}^{\infty }\frac{\omega _k(0;0,r,0)}{n^{2k}}\right) ^{\frac{n}{r}}, \end{aligned}$$
(5.1)

as \(n\rightarrow \infty \), where \((\omega _k(0;0,r,0))_{k\ge 0}\) is determined by

$$\begin{aligned}&\omega _0(0;0,r,0)=1,\nonumber \\&\omega _k(0;0,r,0)=\frac{r}{2k}\sum _{j=0}^{k-1}\frac{B_{2k-2j}}{2k-2j-1} \omega _j(0;0,r,0),\quad k\ge 1. \end{aligned}$$
(5.2)

Corollary 5.4

Let \(r\ne 0\) be a real number. Then

$$\begin{aligned} \prod _{k=1}^nk^{k^2}\sim A_2\cdot n^{\frac{n^3}{3}+\frac{n^2}{2}+\frac{n}{6}} \mathrm {e}^{-\frac{n^3}{9}+\frac{n}{12}} \left( \sum _{k=0}^{\infty }\frac{\omega _k(2;0,r,0)}{n^{2k}}\right) ^{\frac{n}{r}}, \end{aligned}$$
(5.3)

as \(n\rightarrow \infty \), where \((\omega _k(2;0,r,0))_{k\ge 0}\) is determined by

$$\begin{aligned}&\omega _0(2;0,r,0)=1,\nonumber \\&\omega _k(2;0,r,0)=\frac{r}{k}\sum _{j=0}^{k-1}\frac{B_{2k-2j+2}}{(2k-2j+2)(2k-2j+1)(2k-2j-1)} \omega _j(2;0,r,0),\quad k\ge 1. \end{aligned}$$
(5.4)

The first few terms of \((\omega _k(0;0,r,0))\) are

$$\begin{aligned}&\omega _0(0;0,r,0)=1,\quad \omega _1(0;0,r,0)=\frac{r}{12},\quad \omega _2(0;0,r,0)=\frac{r(-4+5r)}{1440},\\&\omega _3(0;0,r,0)=\frac{r(288-84r+35r^2)}{362880},\\&\omega _4(0;0,r,0)=\frac{r(-51840+6096r-840r^2+175r^3)}{87091200}, \end{aligned}$$

and the first few terms of \((\omega _k(2;0,r,0))\) are

$$\begin{aligned}&\omega _0(2;0,r,0)=1,\quad \omega _1(2;0,r,0)=-\frac{r}{360},\quad \omega _2(2;0,r,0)=\frac{r(240+7r)}{1814400},\\&\omega _3(2;0,r,0)=-\frac{r(77760+720r+7r^2)}{1959552000},\\&\omega _4(2;0,r,0)=\frac{r(6531840000+25850880r+110880r^2+539r^3)}{217275125760000}. \end{aligned}$$

Example 5.1

Setting \(r=1\) in Corollary 5.3 gives

$$\begin{aligned} n!&\sim \sqrt{2\pi n}\left( \frac{n}{\mathrm {e}}\right) ^n \left( 1+\frac{1}{12n^2}+\frac{1}{1440n^4}+\frac{239}{362880n^6} -\frac{46409}{87091200n^8}\right. \\&\quad \left. +\frac{9113897}{11496038400n^{10}} -\frac{695818219549}{376610217984000n^{12}}+\cdots \right) ^n,\quad n\rightarrow \infty , \end{aligned}$$

which is the Nemes formula [32]. Besides the case \(r=1\), Nemes gave the case \(r=\frac{4}{5}\) in [31], and Chen presented the case \(r=2\) in [8]. In 2016, Wang obtained the general asymptotic expansion (5.1) and gave the recurrence for the coefficient sequence in [37, Corollary 3.5].

Example 5.2

Setting \(r=1\) in Corollary 5.4 yields

$$\begin{aligned} \prod _{k=1}^nk^{k^2}&\sim A_2\cdot n^{\frac{n^3}{3}+\frac{n^2}{2}+\frac{n}{6}} \mathrm {e}^{-\frac{n^3}{9}+\frac{n}{12}} \left( 1-\frac{1}{360n^2}+\frac{247}{1814400n^4}-\frac{78487}{1959552000n^6}\right. \\&\quad \left. +\frac{6557802299}{217275125760000n^8} -\frac{31014318613001}{726319706112000000n^{10}}+\cdots \right) ^n,\quad n\rightarrow \infty . \end{aligned}$$

Other special cases can be obtained similarly.

6 Conclusions

In this paper, we establish two general asymptotic expansions on the hyperfactorial functions \(\prod _{k=1}^nk^{k^q}\) and the generalized Glaisher–Kinkelin constants \(A_q\). From these two general expansions, we can not only rediscover some asymptotic expansions that have recently appeared in the literature but also obtain new ones. It would be interesting to find more properties of the hyperfactorial functions and the generalized Glaisher–Kinkelin constants by such a unified way.