Suwa, Masanori

Yoshino, Kunio

Kyoto University Research Information Repository

Title Plana's Summation Formula for Holomorphic Functions ofExponential Type (Microlocal Analysis and Related Topics)Author(s)Yo hino, Kunio; Suwa, MasanoriCitation数理解析研究所講究録 (2000), 1158: 180-189Issue Date2000-06URL http://hdl.handle.net/2433/64185RightType Departmental Bulletin PaperTextversionpublisherKyoto University$\mathrm{P}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{a}\mathrm{S})$ Summation Formula for HolomorphicFunctions of Exponential Type(Kunio Yoshino)(Masanori Suwa)Department of Mathematics, Sophia UniversityAbstractIn this paper we derive Plana’s summation formula for holomorphicfunctions of exponential type by using theory of analytic functionalswith unbounded carrier.In \S 1, we will recall the definitions and properties of analytic func-tionals with unbounded carrier.In \S 2, we derive Plana’s summation formula for holomorphic func-tions of exponential type.In \S 3, we will give some examples of Plana’s summation formula.1 The definitions of analytic functionals withunbounded carrier and their properties.Let $L$ and $L_{\epsilon}$ be following strip regions:$L$ $=$ $(-\infty, a]+t[-b, b]$ ,For $\epsilon>0$ , $\acute{L}_{\epsilon}$ $=$ $(-\infty, a+\epsilon]+\iota[-b-\epsilon, b+\epsilon]$ .For $\epsilon>0,$ $\epsilon’>0$ and $k’\in \mathbb{R}$, we put$Q_{b}(L_{\epsilon} : k’+\epsilon’)$$:=$ $\{f(\zeta)\in \mathcal{H}(L_{\mathcal{E}})\cap c(L_{\mathcal{E}}\circ)$ : $\sup_{\zeta\in L\epsilon}|f(\zeta)e^{k\xi}|;+\epsilon’|\xi|<\infty$ , $\zeta=\xi+\iota\eta\}$ .1158 2000 180-189 180$\mathcal{H}(L_{\epsilon}\circ)$ is the space of llolomorphic functions defined on $L_{\epsilon}\circ$ , (interior of $L_{\epsilon}$ ).$C(L_{\epsilon})$ is the space of continuous functions defined on $L_{\epsilon}$ . We put$Q(L:k’)= \lim_{\epsilonarrow 0,\epsilon^{\prime_{arrow}}0}Qb(arrow\xi’L : k\epsilon/+)$ ,where $\lim_{arrow}$ means inductive limit. If $z\in(-k’, \infty)+\iota \mathbb{R}$, then the function $e^{\zeta z}$of $\zeta$ belongs to $Q(L:k’)$ . We denote by $Q’(L:k’)$ the dual space of $Q(L:k’)$ .The elements of $Q’(L : k’)$ is called analytic functionals with carrier $L$ andof type $k’$ .We define the Fourier-Borel transform $\tilde{T}(z)$ of $T\in Q’(L : k’)$ as follows:$\tilde{T}(z)=<T_{\zeta},$ $e^{\zeta}>z$ .$\tilde{T}(z)$ is holomorphic function on the right half plane $(-k’, \infty)+\mathrm{z}\mathbb{R}$ and sat-isfies following estimate:$\forall\epsilon>0,$ $\epsilon’>0,$ $\exists c_{\epsilon,\in}’\geq 0$ such that$|\tilde{T}(Z)|\leq c_{\epsilon,\epsilon}\prime e^{ax}+b|y|+\epsilon|z|$ , $({\rm Re} z\geq-k’+\epsilon’, z=x+\iota y)$ . (1)$\mathrm{E}\mathrm{x}\mathrm{p}((-k’, \infty)+\iota \mathbb{R}$ : $L$ ) denotes the space of holomorphic functions definedon the right half plane $(-k’, \infty)+\iota \mathbb{R}$ and satisfy the estimates (1). Followingtheorem characterizes the Fourier-Borel transform of $Q’(L:k;)$ .Theorem 1.,1 ([3]). Fourier-Borel transform is a linear topological isomor-phism from $Q’(L:k’)$ onto $\mathrm{E}\mathrm{x}\mathrm{p}((-k’, \infty)+\mathrm{z}\mathbb{R}$ : $L$).2 Plana’s summation formula for holomor-phic functions of exponential type.In this section, we will derive Plana’s summation formula for holomorphicfunctions of exponential type by using the theory of analytic functionals withunbounded carrier.181Proposition 2.1. Let $T\in Q’(L:k’)$ . If $k’>0,0\leq b<2\pi,$ ${\rm Res}>a$ and$|{\rm Im} s|+b<2\pi$ , then we have$\sum_{n=0}^{\infty}\tilde{\tau}(n)e-Sn$$= \frac{1}{2}\tilde{T}(\mathrm{o})+\int_{0}^{\infty}\tilde{T}(x)e^{-}dxSx+0\int_{0}\infty\frac{\tilde{T}(lX)e-t\mathit{8}x-\tilde{\tau}(-\iota X)elsx}{e^{2\pi x}-1}d_{X}$ .In order to $1$) $\mathrm{r}\mathrm{o}\mathrm{v}\mathrm{e}$ proposition 2.1, we prepare the following lemmas.Lemma 2.2. Assume that ${\rm Res}>a$ . Then the function $\frac{1}{1-e^{\zeta-s}}$ of $\zeta$ belongsto $Q(L:k’)$ and$\lim_{Narrow\infty}\sum^{N}e-S)n=(\zeta\frac{1}{1-e^{\zeta-s}}n=0$ in $Q(L:k’)$ .Lemma 2.3. $If|{\rm Im}(\zeta-s)|<2\pi$ , we have$\frac{1}{e^{\zeta-s}-1}=\frac{-1}{\zeta-s}-\frac{1}{2}+2\int_{0}^{\infty}\frac{\sin((-S)_{X}}{e^{2\pi x}-1}dX$.Proof. Following equality is well known:$e^{\zeta-s}-11$ $=$ $\frac{1}{2\iota}\sum_{n=1}^{\infty}\{\frac{1}{2\pi n-\iota(\zeta-s)}-\frac{1}{2\pi n+t(\zeta-S)}\}$ .Therefore we have$e^{\zeta-s}-11$ $=$ $\frac{1}{2\iota}\sum_{n=1}^{\infty}\{\frac{1}{2\pi n-\iota(\zeta-s)}-\frac{1}{2\pi n+\mathrm{t}(\zeta-S)}\}$$=$ $\frac{1}{2\iota}\sum_{n=1}^{\infty}\int_{0}^{\infty}e^{-}\{2\pi n-l(\zeta-s)\}x-e^{-}\{2\pi n+\iota(\zeta-s)\}x_{d_{X}}$$=$ $\lim_{Narrow\infty}\int 0\zeta\sum_{n=1}^{N}e^{-}\mathrm{s}\mathrm{i}2\pi nx\mathrm{n}(-S)Xd_{X}\infty$ . (2)182Proof. By proposition 2.4 and Fubini’s theorem, we obtain the lemma2.5.Lemma 2.6.$\langle T_{\zeta},$ $\int_{0}^{\infty}\frac{e^{-l((\mathit{8}}-)x-el(\zeta-s)x}{e^{2\pi x}-1}dx\rangle$ $=$ $\int_{0}^{\infty}\frac{\tilde{T}(-lx)e-^{\tilde{\tau}}lsx(lx)e^{-lS}x}{e^{2\pi x}-1}dX$ .Proof. By proposition 2.4 and Fubini’s theorem, we obtain the lemma2.6.Proof of Proposition 2.1. Since $\tilde{T}(n)=<T_{(},$ $e^{\zeta n}>$ ,$\sum_{n=0}^{\infty}\tilde{\tau}(n)e-Sn$ $=$ $\sum_{n=0}^{\infty}.<\tau\zeta,$ $e>e\zeta n-Sn$$=$ $\langle T_{\zeta},$ $\frac{1}{1-e^{\zeta-s}}\rangle$ , (3)By lemma 2.3, we have$1-e^{\zeta-s}1$ $=$ $\frac{1}{2}+\frac{1}{\zeta-s}+\mathrm{z}\int_{0}^{\infty}\frac{e^{-l(\zeta}-s)x-el(\zeta-s)x}{e^{2\pi x}-1}dx$.By lemma 2.6, we have(3) $=$ $\langle T_{\zeta},$ $\frac{1}{2}+\frac{1}{\zeta-s}+\iota\int 0\frac{e^{-l(\zeta-}s)x-el(\zeta-s)x}{e^{2\pi x}-1}\infty d_{X\rangle}$$=$ $\langle T_{\zeta},$ $\frac{1}{2}\rangle+\langle T_{\zeta},$ $\frac{1}{\zeta-s}\rangle$$+ \iota\int_{0}\infty\{\frac{<T_{\zeta},e-l(\zeta-s)x>-<\tau_{\zeta},el(\zeta-S)x>}{e^{2\pi x}-1}\}dx$$=$ $\frac{1}{2}\tilde{T}(\mathrm{o})+\int_{0}^{\infty}\tilde{\tau}(_{X})e^{-sx_{dX}}$$+ \iota\int_{0}^{\infty}\{\frac{<T_{\zeta},e^{-}l(\zeta-S)x>-<\tau\zeta,et(\zeta-S)x>}{e^{2\pi x}-1}\}dx$ .$=$ $\frac{1}{2}\tilde{T}(\mathrm{o})+\int_{0}^{\infty}\tilde{T}(_{X})e-sxdX+\iota\int_{0}\infty\frac{\tilde{T}(-\iota x)elSx-\tilde{\tau}(xX)e^{-lS}x}{e^{2\pi x}-1}d_{X}$. 1183We put$f_{N}(x)$ $=$ $\sum_{n=1}^{N}e-2\pi nx\sin(\zeta-S)x$ ,$f(x)$ $=$ $| \frac{\sin(\zeta-S)_{X}}{e^{2\pi x}-1}|$ ,then we have$|f_{N}(X)|$ $\leq$ $|f(x)|$ $(x\geq 0)$ ,$f(x)$ $\in$ $L^{1}([0, \infty))$ .Therefore by Lebesgue’s dominated convergence theorem, we have(2) $=$ $\int_{0}^{\infty}\sum_{n=1}e-2\pi nx\mathrm{i}\mathrm{n}(\zeta-S)\mathrm{s}Xdx\infty$$=$ $\int_{0}^{\infty}\frac{\sin(\zeta-s)X}{e^{2\pi x}-1}dX$ .Proposition 2.4 ([4]). Let $T\in Q’(L : k’)$ . For $\forall\epsilon>0$ and $\forall\epsilon’>0_{f}$ thereexists a finite measure $\mu$ on $L_{\epsilon}$ such that$<T,$ $\varphi>$ $=$ $\int_{L_{\epsilon}}\varphi(\zeta)ek’\xi+\frac{\epsilon’}{2}|\xi|d\mu(\zeta)$ $(\zeta--\xi+\mathrm{t}\eta)$ ,for $\varphi(\zeta)\in Q_{b}(L_{\epsilon} : k’+\epsilon’)$ .If $k’>0$ and $s\not\in L$ , then the function $\frac{1}{\zeta-s}$ of $\zeta$ belongs to $Q(L : k’)$ .So we can define the Cauchy-Hilbert tran.sform $\check{T}(s)$ of $T\in Q’(L : k’)$ asfollows:$\check{T}(s)=\langle T_{\zeta},$ $\frac{1}{\zeta-s}\rangle$ .Lemma 2.5. If ${\rm Res}>a$ , we have$\check{T}(s)=\int_{0}^{\infty}\tilde{\tau}(X)e-x_{dx}s$ .184Theorem 2.7. Suppose that $f(z)\in \mathrm{E}\mathrm{x}\mathrm{p}((-k’, \infty)+\iota \mathbb{R}$ : $L$). If $0\leq b<2\pi$ ,$|{\rm Im} s|+b<2\pi,$ ${\rm Res}>a$ and $k’>0$ , then following equality holds:$\sum_{n=0}^{\infty}f(n)e^{-}\mathit{8}n$$= \frac{1}{2}f(0)+\int_{0}^{\infty}f(x)e^{-}d_{X+}tsx\int_{0}\infty\frac{f(lx)e^{-lsx}-f(-llX)elsx}{e^{2\pi x}-1}d_{X}$ . (4)Proof. By theorem 1.1, there exists $T\in Q’(L : k’)$ such that $f(z)=$$\tilde{T}(z)$ . By proposition 2.1, we obtain (4). 1Corollary 2.8. Suppose that $f(z)$ is a holomorphic function on the righthalf plane $(-k’, \infty)+\iota \mathbb{R},$ $k’>0$ and satisfies$\forall\epsilon’>0,$ $\exists C_{\epsilon’}\geq 0$ , $|f(z)| \leq C_{\mathcal{E}’}\frac{e^{b|{\rm Im} z|}}{(1+|_{X}|)^{m}}$ , $({\rm Re} Z\geq-k’+\mathit{6}’, z=x+ty)$ .If $m>1$ and $0\leq b<2\pi$ , then we have$\sum_{n=0}^{\infty}f(n)=\frac{1}{2}f(0)+\int_{0}^{\infty}f(t)dt+?\int_{0}^{\infty}\frac{f(\iota t)-f(-\iota t)}{e^{2\pi t}-1}dt$ . (5)Proof. From the assumption $m>1,$ $\sum_{n=0}^{\infty}f(n)$ converges absolutely and$f(t)\in L^{1}[0, \infty)$ . So we can apply$\mathrm{L}\mathrm{e}\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{g}\mathrm{u}\mathrm{l}\mathrm{e}$dominated convergence theorem.Letting $sarrow \mathrm{O}$ in (4), we obtain (5).Corollary 2.9. Let $f(z)$ be a holomorphic function and ${\rm Im} f(z)\geq 0$ on$(-k’, \infty)+\iota \mathbb{R},$ $k’>0$ . If ${\rm Res}>0$ and $|{\rm Im} s|<2\pi$ , we have$\sum_{n=0}^{\infty}f(n)e-nS=\frac{1}{2}f(0)+\int_{0}^{\infty}f(_{X})e-xs_{dx}+\iota\int_{0}\infty\frac{f(\iota x)e^{-}\iota sx-f(-\mathrm{t}x)e^{ls}x}{e^{2\pi x}-1}d_{X}$ .Proof. In order to prove corollary 2.9, we use the following lemma:Lemma 2.10 ([5]). Let $f(z)$ be a holomorphic and ${\rm Im} f(z)\geq 0$ on the upperhalf plane $\mathbb{R}+\iota \mathbb{R}^{+}:=\{z\in \mathbb{C} : z=x+\iota y, y>0\}$ . Then it satisfies theestimate$|f(z)| \leq\sqrt{2}|f(\mathrm{t})|\frac{1+|_{Z|^{2}}}{|y|}$ $z\in \mathbb{R}+0\mathbb{R}^{+}$ . (6)185Therefore by (6) and theorem 2.7, we obtain the corollary 2.9. 1Proposition 2.11. Let $T\in Q’(L:k’)$ . If $k’>0,0\leq b<2\pi,$ ${\rm Res}>a$ and$|{\rm Im} s|+b<2\pi$ , then we have$\sum_{n=0}^{\infty}\tilde{\tau}(n)e^{-s}n$$= \frac{1}{2}\tilde{T}(0)+\int_{0}^{\infty}\tilde{T}(x)e^{-}dSxx+\sum^{\infty}(\check{\tau}(s+2\pi \mathrm{t}n)-\check{T}n=1(S-2\pi tn))$.Proof. By proposition 2.1 and lemma 2.5, we obtain proposition 2.11.$\iota$3 Examples and applications.Plana’s summation formula is very useful in the theory of the special func-tions [2], [7]. In this section we will give some examples.Example 3.1. For$\sum_{n=0}^{\infty}f(n)=\frac{1}{2}f(0)+\int_{0}^{\infty}f(t)dt+t\int_{0}^{\infty}\frac{f(\iota t)-f(-\iota t)}{e^{2\pi\iota_{-1}}}dt$ ,we put $f(z)=e^{-Az}$ $({\rm Re} A>0, |{\rm Im} A|<2\pi)$ . Then$\sum_{n=0}^{\infty}$ $e$-An$=$ $\frac{1}{2}+\int_{0}^{\infty}e^{-}Atdt+l\int^{\infty}0\frac{e^{-lAt}-e^{lAt}}{e^{2\pi t}-1}dt$$1-e^{-A}1$ $=$ $\frac{1}{2}+\frac{1}{A}+?\int^{\infty}0\frac{e^{-lA}-\iota elAt}{e^{2\pi t}-1}dt$$=$ $\frac{1}{2}+\frac{1}{A}+2\int_{0}^{\infty}\frac{\sin At}{e^{2\pi t}-1}dt$.Here we have$. \mathit{1}_{0}^{\infty}\frac{e^{-lAlA}\iota_{-e}\iota}{e^{2\pi t}-1}dt$ $=$ $\int_{0}^{\infty}(e^{-l}-e^{\iota})AtAt\sum_{=n1}e\infty-2\pi n\iota_{dt}$$=$ $\sum_{n=1}^{\infty}(\frac{1}{2\pi n+tA}-\frac{1}{2\pi n-\iota A})$ .186Hence we have$1-e^{-A}1$ $=$ $\frac{1}{2}+\frac{1}{A}+\sum_{n=1}^{\infty}(\frac{1}{A-2\iota\pi n}+\frac{1}{A+2\iota\pi n})$ ,Example 3.2. Let $f(z)\equiv 1$ , i.e. $f(z)=<\delta_{\zeta},$ $e^{\zeta z}>$ . Then$\sum_{n=0}^{\infty}f(n)e-Sn=\frac{1}{2}f(0)+\int_{0}^{\infty}f(t)e^{-s_{d}}t+\iota\int_{0}^{\infty}t\frac{f(\iota t)e^{-l}-stf(-tt)elst}{e^{2\pi t}-1}dt$bccomes$\sum_{n=0}^{\infty}e-Sn$ $=$ $\frac{1}{2}+\int_{0}^{\infty}e^{-s_{d+}}t\iota\iota\int_{0}^{\infty}\frac{e^{-lstst}-e^{l}}{e^{2\pi t}-1}dt$$\frac{1}{e^{s}-1}+\frac{1}{2}-\frac{1}{s}$ $=$ $2 \int_{0}^{\infty}\frac{\sin st}{e^{2\pi}-\iota 1}dt$ .Example 3.3 (Schl\"omilch expansion ([6])). Let $f(z)=J_{0}(Za)(J_{0}(Z)$ isBessel function with degree $0,$ $a>0,$ $s>0$). Then we have$\sum_{n=1}^{\infty}J0(na)e-nS$$=$ $\frac{1}{\sqrt{a^{2}+s^{2}}}-\frac{1}{2}+\sum_{n=1}\infty\{\frac{1}{\sqrt{a^{2}+(_{S+2_{t}}\pi n)^{2}}}-\frac{1}{\sqrt{a^{2}+(s-2_{l}\pi n)^{2}}}\}\cdot$ .Proof. We have$\int_{0}^{\infty}f(t)e$-stdt $=$ $\int_{0}^{\infty}J_{\mathrm{o}(t)t}ae^{-S\iota}d$1$\sqrt{s^{2}+a^{2}}$.By proposition 2.11, we obtain example 3.3.187Example 3.4. Let $l\in \mathbb{N}\backslash \{0\}$ and $f(z)=J_{l}(za)(J_{l}(z)$ is Bessel functionwith degree $l,$ $a>0,$ $s>0$). Then we have$\sum_{n=1}^{\infty}J_{l}(na)e^{-S}n$ $=$ $\frac{(-s+\sqrt{a^{2}+s^{2}})^{l}}{a^{l}\sqrt{a^{2}+s^{2}}}$$+$ $\sum_{n=1}^{\infty}$ $\{\frac{(-s-2\pi tn+\sqrt{a^{2}+(_{S+}2\pi \mathrm{t}n)^{2}})^{l}}{a^{l}\sqrt{a^{2}+(_{S+}2\pi tn)^{2}}}$$- \frac{(-s+2\pi\iota n-\sqrt{a^{2}+(s-2\pi tn)2})^{l}}{a^{l}\sqrt{a^{2}+(s-2\pi ln)^{2}}}\}$ .Proof. We have$\int_{0}^{\infty}J_{l(}ta)e^{-S}dtt$ $=$$\frac{(-S+\sqrt{a^{2}+s^{2}})l}{a^{l}\sqrt{a^{2}+s^{2}}}$ .We obtain example 3.4. 1Example 3.5 (Hermite’s Formula for Hurwitz Zeta Function ([2])).Let1$f_{a}(\zeta)$ – $a$ is a positive constant, ${\rm Re} z>1$ .$(\zeta+a)^{z}$’Then we have$\sum_{n=0}^{\infty}\frac{1}{(n+a)^{z}}$$=$ $\frac{1}{2a^{z}}+\frac{a^{1-z}}{z-1}.+2\int^{\infty}0\frac{\sin(z\arctan(\frac{t}{a}))}{(e^{2\pi t}-1)(a2+t2)^{\frac{z}{2}}}dt$ $({\rm Re} z>1)$ .Proof. We have$\iota\int_{0}^{\infty}\frac{(-\iota t+a)^{z}-(tt+a)z}{(e^{2\pi t}-1)(a2+t2)^{z}}dt=$ $2 \int_{0}^{\infty}\frac{\sin(z\arctan(\frac{t}{a}))}{(e^{2\pi t}-1)(a2+t2)^{\frac{z}{2}}}dt$ .Therefore we have example 3.5.188Example 3.6 (Appell’s Function ([2])).$f(()= \frac{b^{\zeta}}{(\zeta+a)^{z}},$ $(a, b\in \mathbb{R}, 0<b<1, {\rm Re} z>1, a>0)$ .Then we have$\sum_{n=0}^{\infty}\frac{b^{n}}{(n+a)^{z}}$$=$ $\frac{1}{2a^{z}}+\int_{00}^{\infty}\frac{b^{t}}{(t+a)^{z}}dt+2\int^{\infty}\frac{\sin(z\arctan(\frac{t}{a})-t\log b)}{(e-2\pi l\mathrm{l})(a2+t2)^{\frac{z}{2}}}dt$ .Proof. We have$\mathrm{z}\int^{\infty}0\frac{f(\iota t)-f(-\iota t)}{e^{2\pi t}-1}dt=2\int 0\frac{\sin(Z\arctan(\frac{t}{a})-t\log b)}{(e^{2\pi t}-\mathrm{l})(a2+t2)^{\frac{z}{2}}}dt\infty$.Therefore we have example 3.6. 1References[1] R.P.Boas : Entire Functions, Academic Press, (1954).[2] A.Erdely : Higher Trancsendental Funtions Vol.1, Mcgrawhill, (1953).[3] M.Morimoto : Analytic functionals with non-compact carriers, TokyoJ.Math.vol.1, (1978), 77-103.[4] P.Sargos and M.Morimoto : Transformation des fonctionnelles analy-tiques \‘a porteurs non compacts, Tokyo J.Math.vol.4,(1981), 457-492.[5] V.S.Vladimirov : Holomorphic functions with nonnegative imaginarypart in a tubular domain. over a cone, Math.USSR Sbornik Vol.8, (1969),125-146.[6] G.N.Watson : Theory of Bessel functions, Cambridge University Press,(1966)[7] E.T.Whittaker and G.N.Watson: Modern Analysis, Cambridge Univer-sity Press, (1927)189

Plana's Summation Formula for Holomorphic Functions of Exponential Type (Microlocal Analysis and Related Topics)

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Plana's Summation Formula for Holomorphic Functions of Exponential Type (Microlocal Analysis and Related Topics)

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