数理解析研究所講究録
第 1898 巻 2014 年 156-173

156

ASYMPTOTIC EXPANSIONS FOR CERTAIN -SERIES,-INTEGRALS, -DIFFERENTIALS AND A FORMULA OF
RAMANUJAN FOR SPECIFIC VALUES OF
$q$

$q$

$q$

$\zeta(s)$MASANORI KATSURADA
DEPARTMENT OF MATHEMATICS, HIYOSHI CAMPUS, KEIO UNIVERSITY(慶應義塾大学・経済学部・日吉数学研究室・桂田昌紀)

ABSTRACT. This is a summarized version of the author’s papers [22] [24] on asymptotic
aspects of the -series of Lambert type, -hypergeometric function,-integrals and qdifferentials. Major portions of the results in these papers are rearranged to state inParts I and II respectively; the first part is devoted to showing intrinsic linkage between
asymptotics of certain -series and a formula of Ramanujan for specific values of the, while several complete asymptotic expansions for multiple
Riemann zeta-function-integrals and -differentials of Thomae-Jackson type are presented in the second part.
$q$

$q$

$q$

$q$

$\zeta(s)$

$q$

$q$

Part I: Asymptotics for -series and Ranmanujan’s formula for$q$$\zeta(s)$1.1. Introduction (I). Throughout the present article, let be a complex parameterwith $|q|<1$ , and the substitution $q=e^{-t}$ will be made as it is needed, where the halfplane
Ois transformed to the unit disk $|q|<1$ . It is the main aim of Part Ito
present intrinsic linkage between asymptotic expansions of certain -series (see $(1.1.6)-$(1.1.8) below) and a formula of Ramanujan for specific values of the Riemann zeta-function$q$${\rm Re} t>$$q$

at odd integers (see (1.1.9) below). This linkage is in fact hidden in Ramanujan’s original
work; however, the introduction of the -series (1.1.2) or (1.1.3) and its treatment based ona Mellin transform technique give us an insight for connecting these two aspects together.
Let and be complex variables, and let and be real parameters with $\alpha>0.$
For our later purposes it is convenient to introduce the generalized Lerch zeta-function
defined by
$q$

$z$

$s$

$\alpha$$\mu$$\Phi(s, \alpha, z)$(1.1.1)$\Phi(s, \alpha, z)=\sum_{n=0}^{\infty}(\alpha+n)^{-s}z^{n}$for all if $|z|<1$ , for ${\rm Re} s>0$ if $|z|=1$ and $z\neq 1$ , and for ${\rm Re} s>1$ if $z=1$ , respectively;
this continues to a meromorphic function over the whole -plane and is one-valued in thecomplex -plane cut along the real axis from 1 $to+\infty$ (cf. [13]). We use the notation $e(\mu)=$
$s$

$s$

$z$

hereafter. Then
reduces to the ordinary Lerch zeta-fUnctionwhen$z=e(\mu)$ , so that $\Phi(s, \alpha, 1)=\zeta(s, \alpha)$ is the Hurwitz zeta-function, $e(\mu)\Phi(s, 1, e(\mu))=$the exponential zeta-function, and so $\Phi(s, 1,1)=\zeta(s)$ the Riemann zeta-function.
We remark that the order of the variables in and above differs from the usual notation,in order to retain notational consistency with other terminology.
$e^{2\pi i\mu}$$\phi(s, \alpha, \mu)$$\Phi(\mathcal{S}, \alpha, z)$$\zeta_{\mu}(\mathcal{S})$$\Phi$$\phi$

2010 Mathematics Subject Classification. Primary llP82; Secondary llM35.
Key words and phrases. multiple -integral, multiple -differential, Mellin-transform, asymptotic ex$q$$q$

pansion, Lerch zeta-function.
A portion of the present investigation was initiated during the author’s academic stay at Mathematisches Institut, Westf\"alische Wilhelms-Universit\"at M\"unster. He would like to express his sincere gratitude
to Professor Christopher Deninger and the institution for their warm hospitality and constant support.
The author was also indebted to Grant-in-Aid for Scientific Research (No. 16540038) from JSPS.

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KATSURADA

Let and be real parameters with $\beta>0$ . The main object of the present paper isthe -series of the form$\beta$

$\nu$

$q$

(1.1.2)$S_{s}(_{\mu,\nu}^{\alpha,\beta};q)=e( \beta v)\sum_{m=0}^{\infty}e((\alpha+m)\mu)q^{(\alpha+m)\beta}\Phi(s, \beta, e(v)q^{\alpha+m})$,

which is rewritten, by changing the order of summations, as a Lambert series form
(1.1.3)$S_{s}(_{\mu,\nu}^{\alpha,\beta};q)=e( \alpha\mu)\sum_{n=0}^{\infty}(\beta+n)^{-s}\frac{e((\beta+n)\nu)q^{\alpha(\beta+n)}}{1-e(\mu)q^{\beta+n}}.$We shall prove complete asymptotic expansions ofregion $|\arg t|<\pi/2$ (see Theorem below). Let as usual$S_{s}(_{\mu,\nu}^{\alpha,\beta};q)$

as $tarrow 0$ in the sectorial$0$$(z;q)_{\infty}= \prod_{m=0}^{\infty}(1-zq^{m}) , (z;q)_{n}=(z;q)_{\infty}/(zq^{n};q)_{\infty}$

for any integer denote -shifted factorials. Our main formula (1.2.3) in particular implies
a complete asymptotic expansion ofas $qarrow 1^{-}$ , and it further allows us totreat the -series
$n$

$q$

$\log(q^{\alpha};q)_{\infty}$$q$(1.1.4)(1.1.5)$F(q)= \sum_{n=0}^{\infty}\frac{q^{n^{2}}}{(q;q)_{n}^{2}},$and$G(q)= \sum_{n=0}^{\infty}\frac{q^{n^{2}}}{(q;q)_{n}}$$H(q)= \sum_{n=0}^{\infty}\frac{q^{n(n+1)}}{(q;q)_{n}}.$These are typical examples of the theta series (in the transformed Eulerian form) whose
asymptotic behaviours near the singularities at the points $q^{k}=1(k=1,2, \ldots)$ were firstconsidered by Ramanujan in his last letter to Hardy (see [38]). Ramanujan showed(1.1.6)$F(q)=( \frac{t}{2\pi})^{1/2}\exp(\frac{\pi^{2}}{6t}-\frac{t}{24})+o(1)$(1.1.7)$G(q)=( \frac{2}{5-\sqrt{5}})^{1/2}\exp(\frac{\pi^{2}}{15t}-\frac{t}{60})+o(1)$(1.1.8)$H(q)=( \frac{2}{5+\sqrt{5}})^{1/2}\exp(\frac{\pi^{2}}{15t}+\frac{11t}{60})+o(1)$,,,as $tarrow+0$ , and similar asymptotic formulae for certain other -series. In conjunction withthis result, (complete) Stirling’s formula for the -gamma function was first establishedby Moak [31], while Ueno and Nishizawa [37] developed their theory on a -analogueof the Hurwitz zeta-function and applied it to rederive the same formula, together withasymptotic expansions of $G(q)$ and $H(q)$ , similar to (1.1.7) and (1.1.8). The study on
asymptotic aspects for more general -series of the typewas initiated by Ramanujan [35, p. 366] [36, p.359], and was further proceeded by Berndt [7][8, Chap. 27]. This direction has recently been systematically explored by McIntosh[25][26][27] and Gordon-McIntosh [17][18], in conjunction with transformation propertiesof the -series. It is to be remarked that the basic tool applied by these authors is theEuler-Maclaurin summation device. The Melhn transform technique, on the other hand,
was applied by Meinardus [29] [30] to derive certain asymptotic formulae for fairly generalclass of partition-type functions. We refer the reader to [2, Chap. 6] for various related
works.
$q$

$q$

$q$

$q$

$q$

$\sum_{n=0}^{\infty}a^{n}q^{bn^{2}+cn}/(q;q)_{n}$158
ASYMPTOTICS FOR -SERIES, -INTEGRALS AND -DIFFERENTIALS
$q$

$q$

$q$

Let $B_{k}(k=0,1,2, \ldots)$ denote the Bernoulli numbers (cf. [13]). Our main theorem also
yields Ramanujan’s famous formula for specific values of the Riemann zeta-function at
odd integers (cf. [5][6]), which asserts, for any integer $n\neq 0,$(1.1.9)$\xi^{-n}\{\frac{1}{2}\zeta(2n+1)+\sum_{l=1}^{\infty}\frac{l^{-2n-1}}{e^{2l\xi}-1}\}+2^{2n}\sum_{k=0}^{n+1}\frac{B_{2n+2-2k}B_{2k}}{(2n+2-2k)!(2k)!}\xi^{n+1-k}(-\eta)^{k}$$=(- \eta)^{-n}\{\frac{1}{2}\zeta(2n+1)+\sum_{l=1}^{\infty}\frac{l^{-2n-1}}{e^{2l\eta}-1}\},$

where and are positive real numbers satisfying
and the finite sum on theleft-hand side is to be regarded as null if $n<-1$ (see Theorem 2 in Section 1.4). It
will later turn out that the excluded case $n=0$ of this formula emerges (in a sense) as
asymptotic expansions of $F(q),$ $G(q)$ and $H(q)$ (see Corollary 1.4 in Section 1.3).
$\xi$$\xi\eta=\pi^{2}$$\eta$

1.2. The main theorem (I). Let
the sequence of rational functions

$x$

and

$y$

be complex variables. Apostol [3] introduced
defined by the Taylor series expansion$\mathcal{B}_{k}(x, y)(k\geq 0)$(1.2.1)$\frac{ze^{xz}}{ye^{z}-1}=\sum_{k=0}^{\infty}\frac{\mathcal{B}_{k}(x,y)}{k!}z^{k}$with $|\arg y|<\pi$ near $z=0$ . The function, which coincides with the usual
Bernoulli polynomial $B_{k}(x)$ if $y=1$ , is a polynomial in of degree at most with$co$ efficients in. Next let
be the gamma function, and $U(a;c;z)$ denote theconfluent hypergeometric function defined by$\mathcal{B}_{k}(x, y)$

$k$

$x$

$\mathbb{Q}(y)$(1.2.2)$\Gamma(s)$$U(a;c;z)= \frac{1}{\Gamma(a)}\int_{0}^{\infty e^{i\varphi}}e^{-zw}w^{a-1}(1+w)^{c-a-1}dw$

for ${\rm Re} a>0$ and $|\arg z+\varphi|<\pi/2$ with any fixed angle, where the path of
integration is taken as a half-line from the origin to(cf. [13]); the domain of isextended to the whole sector $|\arg z|<3\pi/2$ by rotating suitably the path of integrationin (1.2.2).
We now state our main result in Part I.
$\varphi\in(-\pi, \pi)$$\infty e^{i\varphi}$

Theorem
let

. Let

$z$

and be real parameters with $\alpha>0$ and $\beta>0,$ $q=e^{-t}$ , andbe defined by (1.1.2) or (1.1.3). Then for any integer $K\geq 0$ and any
in the sector $|\arg t|<\pi/2$ the formula$0$$\alpha,$$\beta,$

$\mu$

$v$

$S_{s}(\alpha, \beta;\mu, \nu;q)$complex(1.2.3)$t$$S_{s}(_{\mu,\nu}^{\alpha,\beta};q)=e(\alpha\mu+\beta\nu)\mathcal{B}_{0}(\beta, e(\nu))\Gamma(1-s)\phi(1-s, \alpha, \mu)t^{s-1}$$+e( \alpha\mu+\beta\nu)\sum_{k=-1}^{K-1}\frac{(-1)^{k+1}\mathcal{B}_{k+1}(\alpha,e(\mu))}{(k+1)!}\phi(s-k, \beta, \nu)t^{k}$$+R_{s,K}(_{\mu,\nu}^{\alpha,\beta};q)$holds in the region ${\rm Re} s<K+1$ except the points $s=k(k=0,1, \ldots, K)$ , whereis defined by (1.2.1), and the empty sum is to be regarded as null. Hereremainder term satisfying the estimate$\mathcal{B}_{k}(x, y)$$R_{s,K}(_{\mu,\nu}^{\alpha,\beta};q)$(1.2.4)is the$R_{s,K}(_{\mu,\nu}^{\alpha,\beta};q)=O(|t|^{K})$as $tarrow 0$ through $|\arg t|\leq\pi/2-\delta$ with any small $\delta>0$ , in the region ${\rm Re} s<K+1,$
where the implied -constant depends at most on $K,$
and . In particular
$O$

$s,$

$\alpha,$$\beta,$$\mu,$$\nu$$\delta$

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KATSURADA

when $K\geq 1,0<\alpha\leq 1,0<\beta\leq 1,0\leq\mu\leq 1$ and $0\leq\nu\leq 1$ the explicit expressio(1.2.5)

160
ASYMPTOTICS FOR -SERIES, -INTEGRALS AND -DIFFERENTIALS
$q$

$q$

$q$

particular if $K\geq 2$ and $0<\alpha\leq 1$ , the explicit expressions as in (1.2.5) follow for theremainder terms.
Remark A complete asymptotic expansion ofas $qarrow 1^{-}$ was first established by
Moak [31] and later rederived by Ueno-Nishizawa [37] in a slightly different form from thatof (1.3.2). McIntosh [25] [27] proved (1.3.2) for real $t>0$ with the error term $R_{1,K}=O(t^{K})$in a more general situation,$(q^{\alpha};q)_{\infty}$

Corollary 1.1. For any real $\alpha>0$ and any integer $K\geq 1$ the formula(1.3.4)$\log(-q^{\alpha};q)_{\infty}=\frac{\pi^{2}}{12t}-B_{1}(\alpha)\log 2+\frac{1}{4}B_{2}(\alpha)t$$- \sum_{k=2}^{K-1}\frac{(-1)^{k}(2^{k}-1)B_{k}B_{k+1}(\alpha)}{k(k+1)!}t^{k}-R_{1,K}(_{0^{\alpha’ 1}},1/2;q)$holds in $|\arg t|<\pi/2$ , where the remainder term
satisfies the same estimate$K\geq2$$0<\alpha\leq1$
as (1.2.4). In particular if
andthe explicit expression as in (1.2.5)
follows for the remainder term.
$R_{1,K}(_{0^{\alpha’ 1}},1/2;q)$

To describe the subsequent results, the change of the base
(1.3.5)$q=e^{-t}\mapsto e^{-4\pi^{2}/t}=\hat{q}$is frequently applied. Noting the facts(1.3.6)$B_{2h+1}=0, h=1,2, \ldots,$(1.3.7)$B_{k}(1-\alpha)=(-1)^{k}B_{k}(\alpha) , k=0,1,2\ldots$(cf. [13]), we find that every term (with $k\geq 2$ ) of the series in (1.3.2) and (1.3.4) vanishes
when $\alpha=1$ , and hence Theorem further reduces to$0$Corollary 1.2. The following formulae hold:(1.3.8)$\log(q;q)_{\infty}=-\frac{\pi^{2}}{6t}-\frac{1}{2}\log\frac{t}{2\pi}+\frac{t}{24}-\sum_{l=1}^{\infty}l^{-1}\frac{\hat{q}^{f}}{1-q\gamma},$or in exponential form$(q;q)_{\infty}= \sqrt{\frac{2\pi}{t}}\exp(-\frac{\pi^{2}}{6t}+\frac{t}{24})(\hat{q};\hat{q})_{\infty}$(1.3.9);$\log(-q;q)_{\infty}=\frac{\pi^{2}}{12t}-\frac{1}{2}\log 2+\frac{t}{24}-\sum_{l=1}^{\infty}l^{-1}\frac{q^{\triangleleft/2}}{1-q\gamma},$or in exponential form$(-q;q)_{\infty}= \frac{1}{\sqrt{2}}\exp(\frac{\pi^{2}}{12t}+\frac{t}{24})(\hat{q}^{t/2};\hat{q})_{\infty}.$

Remark Formulae (1.3.8) and (1.3.9) are classic; these can be found for e.g., in [4, Chap. 3].
Remark. Formulae (1.3.8) and (1.3.9) both give complete (convergent) asymptotic expansions, since for instance the l-th term of the last infinite series in (1.3.8) is of order$\hat{q}^{\{}/l+O(\hat{q}^{2l})$as$larrow\infty.$

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KATSURADA

It can be observed that the explicit expression (1.2.5) for the remainder term, in certain
specific cases (as in the preceding corollary), further reduces to complete (convergent)
asymptotic expansions as $tarrow 0$ in $|\arg t|<\pi/2$ (see Corollaries 1.3-1.5 below). If one
considers, for instance, the logarithm of the pairingwith $0<\alpha<1,$each term (with $k\geq 2$ ) in its asymptotic series vanishes again by (1.3.6) and (1.3.7).
$\mathbb{R}om(1.2.5)$ and Theorem 1 we canin fact prove:$(q^{\alpha};q)_{\infty}(q^{1-\alpha};q)_{\infty}$

Corollary 1.3. The following formula hold for any real

$0<\mu<1$ :(1.3.10)$\alpha$

and

$\mu$

with $0<\alpha<1$ and$\log\{(q^{\alpha};q)_{\infty}(q^{1-\alpha};q)_{\infty}\}=-\frac{\pi^{2}}{3t}+\log(2\sin\pi\alpha)+\frac{1}{2}B_{2}(\alpha)t$$- \sum_{l=1}^{\infty}l^{-1}\frac{e((1-\alpha)l)q\triangleleft}{1-q\gamma}-\sum_{l=1}^{\infty}l^{-1}\frac{e(\alpha l)\hat{q}^{l}}{1-q\gamma},$or in exponential form$(q^{\alpha};q)_{\infty}(q^{1-\alpha};q)_{\infty}=2( \sin\pi\alpha)\exp\{-\frac{\pi^{2}}{3t}+\frac{1}{2}B_{2}(\alpha)t\}$$\cross(e(1-\alpha)\hat{q};\hat{q})_{\infty}(e(\alpha)\hat{q};\hat{q})_{\infty}$;(1.3.11)$\log\{(e(\mu)q^{\alpha};q)_{\infty}(e(1-\mu)q^{1-\alpha};q)_{\infty}\}=-\{\zeta_{\mu}(2)+\zeta_{1-\mu}(2)\}t^{-1}$$-2 \pi iB_{1}(\alpha)B_{1}(\mu)+\frac{1}{2}B_{2}(\alpha)t$$- \sum_{l=1}^{\infty}l^{-1}\frac{e((1-\alpha)l)\hat{q}^{\mu l}}{1q}-\sum_{l=1}^{\infty}l^{-1}\frac{e(\alpha l)\hat{q}^{\langle 1-\mu)l}}{1-q\gamma},$or in exponential form$(e(\mu)q^{\alpha};q)_{\infty}(e(1-\mu)q^{1-\alpha};q)_{\infty}=\exp[\{\zeta_{\mu}(2)+\zeta_{1-\mu}(2)\}t^{-1}-2\pi iB_{1}(\alpha)B_{1}(\mu)$$+ \frac{1}{2}B_{2}(\alpha)t](e(1-\alpha)\hat{q}^{\mu};q\gamma_{\infty}(e(\alpha)^{\wedge-\mu}q;\hat{q})_{\infty}.$

We can now restate Ramanujan’s asymptotic formula $(1.1.6)-(1.1.8)$ with explicit error
terms. It is known that $F(q)=1/(q;q)_{\infty}$ (cf. [38, pp.57-58], and the famous RogersRamanujan identities assert that
$G(q)= \frac{1}{(q;q^{5})_{\infty}(q^{4};q^{5})_{\infty}}$and$H(q)= \frac{1}{(q^{2};q^{5})_{\infty}(q^{3};q^{5})_{\infty}}$(cf. [2, (7. 1.6) and (7. 1.7)]). Formulae (1.3.8) and (1.3. 10) therefore imply

Corollary 1.4. The following formulae hold for $F(q),$ $G(q)$ and $H(q)$ defined by (1.1.4)
and (1.1.5):(1.3.12)$F(q)=( \frac{t}{2\pi})^{1/2}\exp(\frac{\pi^{2}}{6t}-\frac{t}{24})\frac{1}{(\hat{q;}\hat{q})_{\infty}},$162
ASYMPTOTICS FOR -SERIES, -INTEGRALS AND -DIFFERENTIALS
$q$

$q$

$q$

or in logarithmic form$\log F(q)=\frac{\pi^{2}}{6t}+\frac{1}{2}\log\frac{t}{2\pi}-\frac{t}{24}+\sum_{l=1}^{\infty}l^{-1}\frac{\triangleleft q}{1-q\gamma}.$(1.3.13)$G(q)=( \frac{2}{5-\sqrt{5}})^{1/2}\exp(\frac{\pi^{2}}{15t}-\frac{t}{60})\frac{1}{(e(1/5)^{\eta/5}q;\hat{q}^{1/5})_{\infty}(e(4/5)\hat{q}^{1/5\eta/5};q)_{\infty}},$or in logarithmic form;$\log G(q)=\frac{\pi^{2}}{15t}+\frac{1}{2}\log(\frac{2}{5-\sqrt{5}})-\frac{t}{60}+\sum_{l=1}^{\infty}l^{-1}\frac{e(l/5)\hat{q}^{l/5}}{1-q\gamma/5}+\sum_{l=1}^{\infty}l^{-1}\frac{e(4l/5)\hat{q}^{l/5}}{1q}$(1.3.14)$H(q)=( \frac{2}{5+\sqrt{5}})^{1/2}\exp(\frac{\pi^{2}}{15t}+\frac{11t}{60})\frac{1}{(e(2/5)^{\eta/5\eta/5}q;q)_{\infty}(e(3/5)\hat{q}^{1/5};\hat{q}^{1/5})_{\infty}},$or in logarithmic form$\log H(q)=\frac{\pi^{2}}{15t}+\frac{1}{2}\log(\frac{2}{5+\sqrt{5}})+\frac{11t}{60}+\sum_{l=1}^{\infty}l^{-1}\frac{e(2l/5)\hat{q}^{i/5}}{1q}+\sum_{l=1}^{\infty}$l$-1 \frac{e(3l/5)\hat{q}^{f/5}}{1q}.$

We next mention slightly different type of implications from Theorem 1. To this aim
several necessary terminologies are prepared. The -gamma and -beta functions aredefined respectively by$q$$\Gamma_{q}(\alpha)=\frac{(q;q)_{\infty}}{(q^{\alpha};q)_{\infty}}(1-q)^{1-\alpha}$

and

$q$

$B_{q}( \alpha,\beta)=\frac{\Gamma_{q}(\alpha)\Gamma_{q}(\beta)}{\Gamma_{q}(\alpha+\beta)},$whose limits as $qarrow 1^{-}$ are known to be the ordinary gamma function and the beta
, respectively (cf. [16]). Whilst the basic hyperfunction
geometric function $2\phi_{1}(a, b;c;q, z)$ is defined by$B(\alpha, \beta)=\Gamma(\alpha)\Gamma(\beta)/\Gamma(\alpha+\beta)$$2 \phi_{1}(a, b;c;q, z)=\sum_{n=0}^{\infty}\frac{(a;q)_{n}(b;q)_{n}}{(c;q)_{n}(q;q)_{n}}z^{n}, |z|<1,$

for any complex
and with $c\neq q^{-n}(n=0,1,2, \ldots)$ , whose particular caseandgives a -analogue of Gauss’ hypergeometric function(cf. [16, 1.2]). It is known that the classical Gauss’ and Kummer’s summation formulae$a,$$b=q^{\beta}$$c=q^{\gamma}$$b$$a=q^{\alpha},$$c$${}_{2}F_{1}(\alpha, \beta;\gamma;z)$$q$${}_{2}F_{1}( \alpha, \beta;\gamma;1)=\frac{\Gamma(\gamma)\Gamma(\gamma-\alpha-\beta)}{\Gamma(\gamma-\alpha)\Gamma(\gamma-\beta)},$where ${\rm Re}(\gamma-\alpha-\beta)>0,$ $\gamma\neq-n(n=0,1,2, \ldots)$ , and${}_{2}F_{1}( \alpha, \beta;1+\alpha-\beta;-1)=\frac{\Gamma(1+\alpha-\beta)\Gamma(1+\alpha/2)}{\Gamma(1+\alpha)\Gamma(1+\alpha/2-\beta)},$where $1+\alpha-\beta\neq-n(n=0,1,2, \ldots)$ , have -analogues of the form$q$

$2 \phi_{1}(q^{\alpha}, q^{\beta};q^{\gamma};q, q^{\gamma-\alpha-\beta})=\frac{(q^{\gamma-\alpha};q)_{\infty}(q^{\gamma-\beta};q)_{\infty}}{(q^{\gamma};q)_{\infty}(q^{\gamma-\alpha-\beta};q)_{\infty}},$$2 \phi_{1}(q^{\alpha}, q^{\beta};q^{1+\alpha-\beta};q, -q^{1-\beta})=\frac{(-q;q)_{\infty}(q^{1+\alpha};q^{2})_{\infty}(q^{2+\alpha-2\beta};q^{2})_{\infty}}{(q^{1+\alpha-\beta};q)_{\infty}(-q^{1-\beta};q)_{\infty}}$

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respectively (cf. [16, 1.5; 1.8]). Combining formulae (1.3.2) and (1.3.4) with appropriate
exponents (in place of ) we can prove$\alpha$Corollary 1.5. Let
be positive real numbers. Then the following formulae holdfor any integer $K\geq 1$ when $tarrow 0$ through $|\arg t|\leq\pi/2-\delta$ with any small:$\alpha,$$\beta,$$\gamma$$\delta>0$$\log\Gamma_{q}(\alpha)=\log\Gamma(\alpha)-\frac{1}{4}(\alpha-1)(\alpha-2)t$$+ \sum_{k=2}^{K-1}\frac{B_{k}}{kk!}\{\frac{(-1)^{k}B_{k+1}(\alpha)}{k+1}+1-\alpha\}t^{k}+O(|t|^{K})$for$\alpha>0$;$\log B_{q}(\alpha, \beta)=\log B(\alpha, \beta)+\frac{1}{2}(\alpha\beta-1)t$$+ \sum_{k=2}^{K-1}\frac{B_{k}}{kk!}\{\frac{(-1)^{k}C_{k+1}(\alpha,\beta)}{k+1}+1\}t^{k}+O(|t|^{K})$for$\alpha>0$

and $\beta>0$ , where$C_{k}(\alpha, \beta)=B_{k}(\alpha)+B_{k}(\beta)-B_{k}(\alpha+\beta)$;$\log_{2}\phi_{1}(q^{\alpha}, q^{\beta};q^{\gamma};q, q^{\gamma-\alpha-\beta})=\log {}_{2}F_{1}(\alpha, \beta;\gamma;1)-\frac{1}{2}\alpha\beta t$$- \sum_{k=2}^{K-1}\frac{(-1)^{k}B_{k}D_{k+1}(\alpha,\beta,\gamma)}{k(k+1)!}t^{k}+O(|t|^{K})$for$\gamma-\alpha>0,$$\gamma-\beta>0,$ $\gamma>0$

and $\gamma-\alpha-\beta>0$ , where$D_{k}(\alpha,\beta, \gamma)=B_{k}(\gamma-\alpha)+B_{k}(\gamma-\beta)-B_{k}(\gamma)-B_{k}(\gamma-\alpha-\beta)$;$\log_{2}\phi_{1}(q^{\alpha}, q^{\beta};q^{1+\alpha-\beta};q, -q^{1-\beta})=\log{}_{2}F_{1}(\alpha, \beta;1+\alpha-\beta;-1)$$- \sum_{k=2}^{K-1}\frac{(-1)^{k}B_{k}E_{k+1}(\alpha,\beta)}{k(k+1)!}t^{k}+O(|t|^{K})$for $1+\alpha>0,2+\alpha-2\beta>0,1+\alpha-\beta>0$ and $1-\beta>0$ , where$E_{k}(\alpha,\beta)=2^{k-1}B_{k}(\alpha/2+1/2)+2^{k-1}B_{k}(1+\alpha/2-\beta)$$-B_{k}(1+\alpha-\beta)-(2^{k-1}-1)B_{k}(1-\beta)$

Here the implied$O$-constants depend at most on $K,$$\alpha,$$\beta,$$\gamma$and.

$\delta.$

1.4. Connections with Ramanujan’s formula for $\zeta(2n+1)$ . We next describe that
our main theorem implies Ramanujan’s formula for $\zeta(2n+1)$ and its several variants.
In order to clarify symmetricity of the following results we introduce the new parameter$\tau=t/2\pi$ . Then the case $\alpha=\beta=1,$ $\lambda=\mu=0$
and $s=2n+1(n=\pm 1, \pm 2, \ldots)$ of
Theorem reduces to the following equivalent form of (1.1.9).
$0$

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ASYMPTOTICS FOR -SERIES, -INTEGRALS AND -DIFFERENTIALS
$q$

$q$

Theorem 2 (Ramanujan). Let $q=e^{-2\pi\tau}$ andinteger $n\neq 0$ the formula(1.4.1)$q$$\hat{q}=e^{-2\pi/\tau}$with ${\rm Re}\tau>0$ . Then for any$S_{2n+1}(_{0,0}^{1,1};q)+ \frac{1}{2}\zeta(2n+1)+\frac{1}{2}(2\pi)^{2n+1}\sum_{k=0}^{n+1}\frac{(-1)^{k}B_{2n+2-2k}B_{2k}}{(2n+2-2k)!(2k)!}\tau^{2n+1-2k}$$=(-1)_{\mathcal{T}^{2n}\{0,0}^{n}S_{2n+1}(^{1,1}; \hat{q})+\frac{1}{2}\zeta(2n+1)\}$

holds.
Theorem$0$further yields the following several variants of (1.1.9).

Theorem 3. Let and be as in Theorem 2. Then the following formulae hold for any
integer and any real and with $0<\alpha<1$ and $0<\mu<1$ :$\hat{q}$

$q$

$n$

(1.4.2)$\alpha$$\mu$$S_{2n+1}(_{0,\mu}^{\alpha,1};q)+S_{2n+1}(_{0,1-\mu}^{1-\alpha,1};q)+(2 \pi)^{2n+1}\sum_{k=0}^{2n+2}(-i)^{k}B_{2n+2-k}(\alpha)B_{k}(\mu)\tau^{2n+1-k}$$(2n+2-k)!k!$$=(-1)^{n}\mathcal{T}^{2n}\{1-\alpha 0,\alpha\}$ ;(1.4.3)$S_{2n}(_{0,\mu}^{\alpha,1};q)-S_{2n}(_{0,1-\mu}^{1-\alpha,1};q)-(2 \pi)^{2n}\sum_{k=0}^{2n+1}\frac{(-i)^{k}B_{2n+1-k}(\alpha)B_{k}(\mu)}{(2n+1-k)!k!}\tau^{2n-k}$$=i(-1)^{n}\mathcal{T}^{2n-1}\{1-\alpha 0,\alpha\},$where $B_{k}(x)$ denotes the k-th Bernoulli polynomial.
Remark. Eie and Chen [12] recently obtained the same formula as (1.4.2) in a quite
different manner, basing on their theorems for multiple zeta functions associated withpolynomials.

Theorem 4. Let and be as in Theorem 2. Then the following formulae hold for any
integer and any real and with $0<\beta<1$ and $0<\lambda<1$ :$\hat{q}$

$q$

$n$

(1.4.4)$\beta$$\lambda$$S_{2n+1}(_{\lambda,0}^{1,\beta};q)+S_{2n+1}(_{1-\lambda,0}^{1,1-\beta};q)+\zeta(2n+1, \beta)$$+(2 \pi)^{2n+1}\sum_{k=0}^{2n+2_{i^{k}\mathcal{B}_{2n+2-k}(0,e(\lambda))\mathcal{B}_{k}(0,e(\beta))_{\mathcal{T}^{2n+1-k}}}}$$(2n+2-k)!k!$$=(-1)_{\mathcal{T}^{2n}\{\beta,0}^{n}S_{2n+1}(_{1-\beta,0}1,\lambda;\hat{q})+S_{2n+1}(^{1,1\lambda};\hat{q})+\zeta(2n+1,1-\lambda)\}$except when $n=0$ ;(1.4.5)$S_{2n}(_{\lambda_{)}0}^{1,\beta};q)-S_{2n}(_{1-\lambda,0}^{1,1-\beta};q)+\zeta(2n, \beta)$$-(2 \pi)^{2n}\sum_{k=0}^{2n+1}\frac{i^{k}\mathcal{B}_{2n+1-k}(0,e(\lambda))\mathcal{B}_{k}(0,e(\beta))}{(2n+1-k)!k!}\tau^{2n-k}$$=i(-1)^{n}\tau^{2n-1}\{1,\lambda\beta,0,$
where$\mathcal{B}_{k}(x, y)$is defined by (1.2.1).

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Part II: Asymptotics for multiple -integrals and -differentials
2.1. Introduction (II). Suppose temporarily that is a real parameter with $0<q<1.$
Let
be a function integrable on the interval $[0, x].$-analogue of the ordinary
integral, in the form
$q$

$q$

$q$

$\varphi(u)$

$A$

$q$

$\int_{0}^{x}\varphi(u)du$(2.1.1)$\int_{0}^{x}\varphi(u)d_{q}u=(1-q)x\sum_{n=0}^{\infty}\varphi(q^{n}x)q^{n},$was introduced by Thomae [34] in 1869 and studied by Jackson [19] during 1910-1951(see also [16, p.23, Chap.1, 1.11]). The formulation in (2.1.1) is motivated from the factthat

(2.1.2)$q arrow 1^{-}hm\int_{0}^{x}\varphi(u)d_{q}u=\int_{0}^{x}\varphi(u)du$holds for allcontinuous on $[0, x]$ . On the other hand, a -analogue of the ordinary
differentiation is formulated as$\varphi(u)$$q$(2.1.3)$\partial_{q,z}\psi(z)=\frac{\psi(z)-\psi(qz)}{(1-q)z}$(cf. [16, p.27, 1.12]), which asserts that
(2.1.4)$\lim_{qarrow 1^{-}}\partial_{q,z}\psi(z)=\psi’(z)=\partial_{z}\psi(z)$,say, for allcomplex differentiable atThroughout the following, is a complex parameter with $0<|q|<1$ , and the substitution $q=e^{-t}$ will be made if necessary, upon transforming the half-plane ${\rm Re} t>0$ to
the unit disk $|q|<1.$complex domain
is called star-shaped if $0\in D$ and for
any $z\in D$ the line segment
is included in $D$ . We suppose throughout that $f(z)$ is afunction holomorphic in a star-shaped domain $D$ , anddenotes the distance between
and the singularity of $f(z)$ being closest to
We introduce the -integral and -differential operators
anddefined for any
real $x>0$ and $y\geq 0$ by$\psi(z)$

$z.$

$q$

$A$

$D\subset \mathbb{C}$$\overline{0,z}$$0$$\rho_{f}$

$0.$

$q$

$q$

$\mathcal{I}_{q,z}^{x}$$\mathcal{D}_{q,z}^{y}$(2.1.5)$\mathcal{I}_{q,z}^{x}f(z)=\int_{0}^{1}u^{x-1}f(uz)d_{q}u=z^{-x}\int_{0}^{z}w^{x-1}f(w)d_{q}w,$(2.1.6)$\mathcal{D}_{q,z}^{y}f(z)=\frac{f(z)-q^{y}f(qz)}{1-q}=z^{-y}(z\partial_{q,z})\{z^{y}f(z)\}$for any in$z$$|z|<\rho_{f\rangle}$

where the latter equalities follow from (2.1.1) and (2.1.3) respectively.

Remark. If the base is restricted to the range $0<q<1$ , then the domain of inwhich the definitions in (2.1.5) and (2.1.6) are valid is extended to the wholeby itsstar-shapedness.
$q$

$z$

$D$

Proposition 1. The operator relations$\mathcal{I}_{q,z}^{x}\mathcal{D}_{q,z}^{x}=1$and$\mathcal{D}_{q,z}^{x}\mathcal{I}_{q,z}^{x}=1$hold for any $x>0$ , where 1 denotes the identity operation.
It is the main aim of Part II to pursue the directions in (2.1.2) and (2.1.4) further;
this leads us to show that complete asymptotic expansions as $tarrow 0$ through the sector

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ASYMPTOTICS FOR -SERIES, -INTEGRALS AND -DIFFERENTIALS
$q$

$q$

$q$

$|\arg t|<\pi/2$ exist for the multiple -integrals(Theorem 5) and the multiple -differentials(Theorem 6) with any integer $r\geq 1$ , under fairly genericsituations.
full extension of the domain of in which Theorems 5 and 6 are vahd ispossible if $0<q<1$ (Theorem 7). Several applications of our main formulae (2.2.4)
and (2.2.9) will further be given for the Hurzitz-Lerch zeta-function (Theorems 8 and 9),-factorials (Corollary 8.1), and -analogues of the exponential functions (Corollary 8.2),of the binomial functions (Corollary 8.3), and of the poly-logarithmic functions (Corollaries 8.4 and 9.1). As for methodology, it is fundamental to apply a Mellin transform
technique in the proofs of Theorems 5 and 6.
$(\mathcal{I}_{q,z}^{x})^{r}f(q^{y}z)$$q$$(\mathcal{D}_{q,z}^{x})^{r}f(q^{y}z)$

$q$

$A$

$z$

$q$

$q$

2.2. The main theorems (II). Let be any integer, and a complex variable. To describe our results we introduce the functions $A_{f,k}(x, z)$ and N\"orlund’s generalized Bernoulli
by the Taylor
polynomialsof rank (cf. [32]) defined respectively for $k=0,1,$
series expansions
$w$

$r$

$B_{k}^{(r)}(y)$$r$$\ldots$(2.2.1)$e^{xw}f(e^{w}z)= \sum_{k=0}^{\infty}\frac{A_{f,k}(x,z)}{k!}w^{k},$(2.2.2)$e^{yw}( \frac{w}{e^{w}-1})^{r}=\sum_{k=0}^{\infty}\frac{B_{k}^{(r)}(y)}{k!}w^{k}$near $w=0$ . Note that $B_{k}^{(1)}(y)=B_{k}(y)$ is the usual Bernoulli polynomial, and so $B_{k}(0)=$, and use Euler’s differential
is the usual Bernoulh number. We write
operator
We state our first main result in Part II.
$B_{k}^{(r)}(0)=B_{k}^{(r)}$$B_{k}$$\theta_{z}=z\partial_{z}.$Theorem 5. Let and be real parameters with $x>0$ and $y\geq 0,$ $q=e^{-t}$ , and $r\geq 1$
an arbitrary fixed integer. Further let
denote the -times iterated operation$f(z)$(2.1.5)holomorphic
star-shapedanyadomain , and define thetoinoffunction$A_{f,-j}(x,z)(j=1,2,\ldots)$bycoefficients
$x$

$y$

$(\mathcal{I}_{q,z}^{x})^{r}f(z)$

$r$

$D$

(2.2.3)$A_{f,-j}(x, z)= \int_{0}^{1}u_{j}^{x-1}\int_{0}^{1}u_{j-1}^{x-1}\cdots\int_{0}^{1}u_{1}^{x-1}f(u_{1}\cdots u_{j}z)du_{1}\cdots du_{j}.$Then for any integer $K\geq 0$ the formula(2.2.4)$\frac{q^{xy}}{(1-q)^{r}}(\mathcal{I}_{q,z}^{x})^{r}f(q^{y}z)=\sum_{j=1}^{r}\frac{(-1)^{r-j}A_{f,-j}(x,z)B_{r-j}^{(r)}(y)}{(r-j)!}t^{-j}$$+ \sum_{k=0}^{K-1}\frac{(-1)^{r+k}A_{f,k}(x,z)B_{r+k}^{(r)}(y)}{(r+k)!}t^{k}+R_{f,K}^{(r)}(x, y;q, z)$

is the remainderholds in the sector $|\arg t|<\pi/2$ and on the disk $|z|<\rho_{f}$ . Heretransform,
term expressed by a certain inverse Mellinand satisfies the estimate$R_{f,K}^{(r)}$(2.2.5)$R_{f,K}^{(r)}(x, y;q, z)=O(|t|^{K})$as $tarrow 0$ through $|\arg t|\leq\pi/2-\delta$ with any small $\delta>0$ , where the implied -constant$K$ and. In particular if $0\leq y\leq r$ and $K\geq 1$ thedepends at most on
$O$

$r,$

$x,$

$y,$

$z,$

$\delta$

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representation(2.2.6)$R_{f,K}^{(r)}(x, y;q, z)=(-t)^{K} \sum_{l=0}^{r-1}\frac{(-1)^{r-1-l}B_{r-1-l}^{(r)}(y)}{l!(r-1-l)!}\sum_{n=-\infty}^{\infty}\frac{e(ny)}{(2\pi in)^{K+l}}/$$\cross(\frac{\partial}{\partial u})^{l}u^{K+l}\int_{0}^{1}\xi^{xtu+2\pi in-1}(x+\theta_{z})^{K}f(\xi^{tu}z)d\xi|_{u=1}$

follows, where the primed summation symbol indicates that the term with $n=0$ is to beomitted. with $n=0.$
Remark 3. The explicit expression (2.2.6) will be used to extend the domain of(2.2.4) with (2.2.5) is valid (see Theorem 7).

$z$

where

From a point of view of applications it is necessary to establish the asymptotic expansions forboth with and without the associated -multiples (see (2.3.5), (2.3.11)
and (2.3.12) below). The case $y=0$ of Theorem 5 in fact yields, in view of the latter
equality in (2.1.5), the following corollary.
$(\mathcal{I}_{q,z}^{x})^{r}f(z)$Corollary 5.1. Let
asymptotic formula(2.2.7)

$q$

$r$

and

$x$

be as in Theorem 5.

Then for any integer $K\geq 0$ the$\int_{0}^{z}w_{r}^{-1}\int_{0}^{w_{r}}w_{r-1}^{-1}\cdots w_{2}^{-1}\int_{0}^{w}2w_{1}^{x-1}f(w_{1})d_{q}w_{1}\cdots d_{q}w_{r}$$= \sum_{k=0}^{K-1}\frac{(-1)^{k}C_{f,k}(x,z)}{k!}t^{k}+O(|t|^{K})$holds as $tarrow 0$ through $|\arg t|\leq\pi/2-\delta$ for any small $\delta>0$ , on the disk $|z|<\rho_{f}$ with$|\arg z|<\pi$ , where the implied$K$ and-constant depends at most on $x,$
. Here the$C_{f,k}^{(r)}(k=0,1, \ldots)$ are given bycoefficients$O$(2.2.8)$z,$$\delta$$C_{f,k}^{(r)}(x, z)= \sum_{j=\max(1,r-k)}^{r}(\begin{array}{l}kr-j\end{array})B_{k-r+j}^{(-r)}B_{r-j}^{(r)}$$\cross\int_{0}^{z}w_{j}^{-1}\int_{0}^{w_{j}}w_{j-1}^{-1}\cdots w_{2}^{-1}\int_{0}^{w}2w_{1}^{x-1}f(w_{1})dw_{1}\cdots dw_{j}$$+ \sum_{j=0}^{k-r}(\begin{array}{l}kr+j\end{array})B_{k-r-j}^{(-r)}B_{r+j}^{(r)}\theta_{z}^{j}\{z^{x}f(z)\},$

which is reduced if $r=1$ to$C_{f,k}^{(1)}(x, z)= \frac{1}{k+1}[\int_{0}^{z}w^{x-1}f(w)dw+\sum_{j=0}^{k-1}(\begin{array}{ll}k +1j +1\end{array})B_{j+1} \theta_{z}^{j}\{z^{x}f(z)\}],$

where the empty sums are to be regarded as null.

The case $K=1$ of Corollary 5.1 implies the following.

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ASYMPTOTICS FOR -SERIES, -INTEGRALS AND -DIFFERENTIALS
$q$

$q$

$q$

Corollary 5.2. Under the same assumptions as in Corollary 5.1 we have the limiting
relation$|q|<1 \lim_{qarrow 1}\int_{0}^{z}w_{r}^{-1}\int_{0}^{w_{r}}w_{r-1}^{-1}\cdots w_{2}^{-1}\int_{0}^{w}2w_{1}^{x-1}f(w_{1})d_{q}w_{1}\cdots d_{q}w_{r}$$=C_{f,0}^{(r)}(x, z)= \int_{0}^{z}w_{r}^{-1}\int_{0}^{w_{r}}w_{r-1}^{-1}\cdots w_{2}^{-1}\int_{0}^{w2}w_{1}^{x-1}f(w_{1})dw_{1}\cdots dw_{r}.$denote the
We proceed to state our second main result in Part II. For this, let$(s)_{n}=\Gamma(s+n)/\Gamma(s)$gamma function, andfor any integer the rising factorial.
$\Gamma(s)$$n$Theorem 6. Let $x\geq 0$ and $y\geq 0$ be real parameters, $q=e^{-t}$ , and $r\geq 1$ an arbitrarily
denote the -times iterated operation of (2.1.6) to
fixed integer. Further let
any function $f(z)$ holomorphic in a star-shaped domain D. Then for any integer $K\geq 0$
the formula$r$$(\mathcal{D}_{q,z}^{x})^{r}f(z)$(2.2.9)$q^{xy}( \frac{1-q}{t})^{r}(\mathcal{D}_{q,z}^{x})^{r}f(q^{y}z)=\sum_{k=0}^{K-1}\frac{(-1)^{k}A_{f,r+k}(x,z)B_{k}^{(-r)}(y)}{k!}t^{k}+R_{f,K}^{(-r)}(x, y;q, z)$

is the remainderholds in the sector $|\arg t|<\pi/2$ and on the disk $|z|<\rho_{f}$ . Hereterm expressed by a certain inverse Mellin transform, and satisfies the estimate$R_{f,K}^{(-r)}$(2.2.10)$R_{f,K}^{(-r)}(x, y;q, z)=O(|t|^{K})$, where the implied -constantas $tarrow 0$ through $|\arg t|\leq\pi/2-\delta$ with any small$K$
depends at most on
and . Furthermore, for any real $x\geq 0$ and $y\geq 0$ , and$\delta>0$

$r,$

$x,$

$y,$

$O$

$\delta$$z,$

any integer $K\geq 0,$(2.2.11)$R_{f,K}^{(-r)}(x, y;q, z)= \frac{(-1)^{r+K}t^{K}}{\Gamma(r+K)}\sum_{n=0}^{r}\frac{(-r)_{n}}{n!}(y+n)^{r+K}\int_{0}^{1}(1-\xi)^{r+K-1}q^{x(y+n)\xi}$$\cross(x+\theta_{z})^{r+K}f(q^{(y+n)\xi}z)d\xi.$

In view of the latter equality in (2.1.6), the case $y=0$ of Theorem 6 in fact yields thefollowing corollary.
Corollary 6.1. Let
asymptotic formula(2.2.12)

$r$

and

$x$

be as in Theorem 6.

Then for any integer $K\geq 0$ the$(z \partial_{q,z})^{r}\{z^{x}f(z)\}=\sum_{k=0}^{K-1}\frac{(-1)^{k}C_{f,k}^{(-r)}(x,z)}{k!}t^{k}+O(|t|^{K})$holds as $tarrow 0$ through $|\arg t|\leq\pi/2-\delta$ for any small $\delta>0$ , on the disk $|z|<\rho_{f}$ with$|\arg z|<\pi$ , where the implied-constant depends at most on $x,$ $K$ and . Here thecoefficients $C_{f,k}^{(-r)}(k=0,1, \ldots)$ are given by$O$(2.2.13)

$r,$

$z,$

$C_{f,k}^{(-r)}(x, z)= \sum_{j=0}^{k}(\begin{array}{l}kj\end{array})B_{k-j}^{(r)}B_{j}^{(-r)}\theta_{z}^{r+j}\{z^{x}f(z)\},$

which reduces if $r=1$ to$C_{f,k}^{(-1)}(x, z)= \frac{1}{k+1}\sum_{j=0}^{k}(_{j}^{k}I_{1}^{1})B_{k-j}\theta_{z}^{1+j}\{z^{x}f(z)\}.$$\delta$

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The case $K=1$ of Corollary 6.1 implies the following corollary.

Corollary 6.2. Under the same assumptions as in Corollary 6.1 we have the limiting
relation$|^{qarrow 1} \lim_{q|<1}(z\partial_{q,z})^{r}f(z)=C_{f,0}^{(-r)}(x, z)=(z\partial_{z})^{r}\{z^{x}f(z)\}.$We lastly proceed to state the full extension of the domain of in Theorems 5 and 6
under the restriction that $0<q<1$ (see Remark just below of (2.1.6)).
$z$

Theorem 7. Set $q=e^{-t}$ with any real $t>0$ , and let $f(z)$ be any function holomorphic
in a star-shaped domain $D.$
i Let and be real with $x>0$ and $0\leq y\leq r$ . Then the asymptotic expansion(2.2.4) with the estimate (2.2.5) when $tarrow 0^{+}$ , as well as the explicit expression(2.2.6), remain valid throughout the domain $D$ ;ii) Let $x\geq 0$ and $y\geq 0$ be real. Then the asymptotic expansion (2.2.9) with theestimate (2.2.10) when $tarrow 0^{+}$ , as well as the explicit expression (2.11), remain
valid throughout the domain $D$ ;iii) The asymptotic expansion (2.2.7) with (2.2.8) when $tarrow 0^{+}$ for $x>0$ , and also
(2.2.12) with (2.2.13) when $tarrow 0^{+}$ for $x\geq 0$ , remain valid both throughout thedomain $D.$
2.3. Applications of Theorems 5 and 6. We suppose throughout this section that$0<q<1$ . Let $[s]_{q}=(1-q^{S})/(1-q)$ be a -analogue of , and $[s]_{q;n}= \prod_{m=0}^{n-1}[s+m]_{q}$ and$[1]_{q;n}=[n]_{q}!$ for $n=0,1,$
denote -analogues of the rising factorial and the factorialof respectively (cf. [16, p.7, Chap.1]), where the empty products are regarded to be 1.
Note that the limiting relation
implies that
(2.3.1)
and
$x$

$)$

$y$

$s$

$q$

$\ldots$$q$$n$$\lim_{qarrow 1^{-}}[s]_{q}=s$$qarrow 1^{-}hm[s]_{q;n}=(s)_{n}$$\lim_{qarrow 1^{-}}[n]_{q}!=n!.$Recall that the generalized Lerch zeta-function $\Phi(s, x, z)$ is defined by(2.3.2)$\Phi(s, x, z)=\sum_{m=0}^{\infty}(x+m)^{-s}z^{m}$for any complex if $|z|<1$ , and for ${\rm Re} s>1$ if $|z|=1$ (cf. [13]); this is continued to aholomorphic function of, where(2.3.3)$D=\{z\in \mathbb{C}||\arg(1-z)|<\pi\}=\mathbb{C}\backslash [1, +\infty)$
is a complex cut-plane; note here that $D$ is a star-shaped domain. We can therefore apply
the part i) of Theorem 7 (upon (2.2.4) with (2.2.5)) to $f(z)=\Phi(s, x, z)$ , and obtain thefollowing theorem.
$s$$(s, z)\in \mathbb{C}\cross D$

Theorem 8. Let and be real with $x>0$ and $0\leq y\leq r$ , andfor any integer $K\geq 0$ the asymptotic expansion$x$(2.3.4)$y$$s$

any complex. Then$\frac{q^{xy}}{(1-q)^{r}}(\mathcal{I}_{q,z}^{x})^{r}\Phi(\mathcal{S}, x, q^{y}z)=\sum_{j=1}^{r}\frac{(-1)^{r-j}\Phi(s+j,x,z)B_{r-j}^{(r)}(y)}{(r-j)!}t^{arrow}$$+ \sum_{k=0}^{K-1}\frac{(-1)^{r+k}\Phi(s-k,x,z)B_{r+k}^{(r)}(y)}{(r+k)!}t^{k}+O(t^{K})$holds as $tarrow 0^{+}$ , in $|\arg(1-z)|<\pi$ , where the implied
and $K.$

$r,$

$s,$

$x,$

$y,$

$z,$

$O$

-constant depends at most on

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ASYMPTOTICS FOR -SERJES, -INTEGRALS AND -DIFFERENTIALS
$q$

$q$

$q$

Let
for anybe the poly-logarithmic function defined by $Li_{l}(z)=z\Phi(l, 1, z)$
for any $z\in D$ . It is seen from (2.1.1), (2.1.5), (2.1.8) and the relation $\log(1-z)=$$-z\Phi(1,1, z)$ , by (2.3.2), that
$Li_{l}(z)$$l\in \mathbb{Z}$(2.3.5)$\log(q^{y}z;q)_{\infty}=-\frac{q^{y_{Z}}}{1-q}\mathcal{I}_{q,z}^{1}\Phi(1,1, q^{y}z)$

for any real $y\geq 0$ and in $|\arg(1-z)|<\pi$ . Then the case $(r, s, x)=(1,1,1)$ of Theorem 7
yields the following corollary.
Corollary 8.1. Let
expansion(2.3.6)$y$be real with $0\leq y\leq 1$ . Then for any integer $K\geq 0$ the asymptotic$\log(q^{y}z;q)_{\infty}=-Li_{2}(z)t^{-1}-\sum_{k=0}^{K-1}\frac{(-1)^{k+1}Li_{1-k}(z)B_{k+1}(y)}{(k+1)!}t^{k}+O(t^{K})$holds as $tarrow 0^{+}$ , in $|\arg(1-z)|<\pi$ , where the implied
and $K.$

$y,$

$O$

-constant depends at most on$z$

Remark The assertion (2.3.6) was first established by McIntosh [25][27] in a more general
setting.

We next present the applications to -analogues of the exponential and binomial functions defined respectively by$q$,$e_{q}(z)= \sum_{n=0}^{\infty}\frac{z^{n}}{[n]_{q}!} (|z|<\frac{1}{1-q})$$f_{q}(y;z)= \sum_{n=0}^{\infty}\frac{[y]_{q;n}}{[n]_{q}!}z^{n} (|z|<1)$,

from which with (3.1) the limiting relations $\lim_{qarrow 1^{-}}e_{q}(z)=e^{z}$ and $\lim_{qarrow 1^{-}}f_{q}(y;z)=$$(1-z)^{-y}$ follow. It is known that the -binomial theorem (cf. [16, p.8, Chap.l, 1.3])
asserts that$q$(2.3.7)$e_{q}(z)= \frac{1}{((1-q)z;q)_{\infty}}$and$f_{q}(y;z)= \frac{(q^{y}z;q)_{\infty}}{(z;q)_{\infty}}$

for any $y\geq 0$ ; these further provide the meromorphic continuations ofand $f_{q}(y;z)$respectively over the whole -plane.
Corollary 8.1 can therefore be applied to the right sides above on yielding the following
corollaries.
$e_{q}(z)$$z$Corollary 8.2. For any integer $K\geq 0$ the asymptotic expansion(2.3.8)holds as$\log e_{q}(z)=z+\sum_{k=1}^{K-1}\alpha_{k}(z)t^{k}+O(t^{K})$$tarrow 0^{+}$, in $|\arg(1-z)|<\pi$ , and this further implies that$e_{q}(z)=e^{z} \{1+\sum_{k=1}^{K-1}\beta_{k}(z)t^{k}+O(t^{K})\}$

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as$tarrow 0^{+}$, where the coefficients(2.3.9)$\alpha_{k}(z)$and$\beta_{k}(z)$are given by$\alpha_{k}(z)=\sum_{j=0}^{k}\frac{(-1)^{k-j}B_{k-j}}{(k-j)!}\sum_{h=0}^{j}(1+h)^{k-j-2}\frac{B_{j-h}^{(-h-1)_{Z^{1+h}}}}{(j-h)!},$$\beta_{k}(z)=\iota_{j}\geq 0\Sigma_{j=1}^{k}jl_{j}=k\sum_{(j=1,\ldots,k)}\prod_{j=1}^{k}\frac{\alpha_{j}(z)^{l_{j}}}{l_{j}!}$for $k=0,1,$$\ldots$, and the implied $O$ -constants depend on

Corollary 8.3. Let
expansion

$y$

$z$

and $K.$be real with $0\leq y\leq 1$ . Then for any integer $K\geq 0$ the asymptotic$\log f_{q}(y;z)=\sum_{k=0}^{K-1}\frac{(-1)^{k+1}Li_{1-k}(z)}{(k+1)!}\{B_{k+1}-B_{k+1}(y)\}t^{k}+O(t^{K})$holds as$tarrow 0^{+}$, in $|\arg(1-z)|<\pi$ , and this further implies that$f_{q}(y;z)=(1-z)^{-y} \{1+\sum_{k=1}^{K-1}\gamma_{k}(y, z)t^{k}+O(t^{K})\}$

as$tarrow 0^{+}$, where the coefficients$\gamma_{k}(y, z)$

are given by$\gamma_{k}(y, z)=(-1)^{k}\sum_{\Sigma_{j=1}^{k}jl_{j}=k}\prod_{j=1}^{k}\frac{1}{l_{j}!}[\frac{Li_{1-j}(z)}{(j+1)!}\{B_{j+1}(y)-B_{j+1}\}]^{l_{j}}$$\iota_{j}\geq 0(j=1,\ldots,k)$

for $k=0,1,$$\ldots$. Here the implied$O$-constants depend at most on

$y,$

$z$

and $K.$We thirdly present applications to a -analogue of the poly-logarithmic function
for any
defined by$q$$Li_{q,l}(z)$$l\in \mathbb{Z}$(2.3.10)$Li_{q,l}(z)=\sum_{m=0}^{\infty}\frac{z^{1+m}}{[1+m]_{q}^{l}} (|z|<1)$,

which with (2.3.1) asserts that $\lim_{qarrow 1^{-}}Li_{q,l}(z)=Li_{l}(z)$ . We can in fact show
(2.3.11)$Li_{q,r}(z)=z(\mathcal{I}_{q,z}^{1})^{r}\Phi(0,1, z)$

for any integer $r\geq 0$ ; this further provides the meromorphic continuation of $Li_{q,r}(z)$ for
all $z\in D$ . Corollary 5.1 can therefore be applied upon taking $f(z)=\Phi(0,1, z)$ to yield
the following corollary.

Corollary 8.4. Let
the asymptotic expansion$r\in \mathbb{Z}$be arbitrarily fixed with $r\geq 1$ . Then for any integer $K\geq 0$$Li_{q,r}(z)=\sum_{k=0}^{K-1}\frac{(-1)^{k}C_{f,k}^{(r)}(1,z)}{k!}t^{k}+O(t^{K})$172
ASYMPTOTICS FOR -SERIES, -INTEGRALS AND -DIFFERENTIALS
$q$

$q$

holds as$tarrow 0^{+}$$q$, in $|\arg(1-z)|<\pi$ , where the coefficients$C_{f,k}^{(r)}$are given by$C_{f,k}^{(r)}(1, z)= \sum_{j=\max(1r-k)}^{r},(\begin{array}{l}kr-j\end{array})B_{k-r+j}^{(-r)}B_{r-j}^{(r)}Li_{j}(z)$$+ \sum_{j=0}^{k-r}(\begin{array}{l}kr+j\end{array})B_{k-r-j}^{(-r)}B_{r+j}^{(r)}Li_{-j}(z)$for $k=0,1,$. Here the implied$\ldots$$O$-constant depends at most on

$r,$

$z$

and $K.$

We fourthly discuss the applications of Theorem 6; this at first yields on taking $f(z)=$the following theorem.

$\Phi(s, x, z)$Theorem 9. Let $x\geq 0$ and $y\geq 0$ be real, and$K\geq 0$ the asymptotic expansion$s$

any complex. Then for any integer$q^{xy}( \frac{1-q}{t})^{r}(\mathcal{D}_{q,z}^{x})^{r}\Phi(s, x, q^{y}z)=\sum_{k=0}^{K-1}\frac{(-1)^{k}\Phi(s-r-k,x,z)B_{k}^{(-r)}(y)}{k!}t^{k}+O(t^{K})$holds as $tarrow 0^{+}$ , in $|\arg(1-z)|<\pi$ , where the implied
zand $K.$

$r,$

$s,$

$x,$

$O$

-constant depends at most on$y,$

We can in fact show
(2.3.12)$Li_{q,-r}(z)=z(\mathcal{D}_{q,z}^{1})^{r}\Phi(0,1, z)$

for any integer $r\geq 0$ . Corollary 6.1 can therefore be applied by taking $f(z)=\Phi(0,1, z)$
to yield the following corollary.
Corollary 9.1. Let
the asymptotic expansion$r\in \mathbb{Z}$be arbitrarily fixed with $r\geq 1$ . Then for any integer $K\geq 0$$Li_{q,-r}(z)=\sum_{k=0}^{K-1}\frac{(-1)^{k}C_{f,k}^{(-r)}(1,z)}{k!}t^{k}+O(t^{K})$holds as$tarrow 0^{+}$, in $|\arg(1-z)|<\pi$ , where the coefficients$C_{f,k}^{(-r)}$are given by$C_{f,k}^{(-r)}(1, z)= \sum_{j=0}^{k}(\begin{array}{l}kj\end{array})B_{k-j}^{(r)}B_{j}^{(-r)}Li_{-r-j}(z)$for $k=0,1,$$\ldots$. Here the implied$O$-constant depends at most on

$r,$

$z$

and $K.$

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DEPARTMENT OF MATHEMATICS, HIYOSHI CAMPUS, KEIO UNIVERSITY, 4-1-1 HIYOSHI, KOUHOKU, YOKOHAMA 223-8521, JAPAN-mail address: katsurad@z3. keio. jp$E$