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No.	Title	Date of publication	Author(s)	Journal name	Volume/number/page	Publishing type	DOI1	URL1	URL2	Publisher	Description
1	Archimedean zeta integrals for GL(3) × GL(2)	2022/06	Miki Hirano Taku Ishii Tadashi Miyazaki	Memoirs of the American Mathematical Society	278/ 1366, 1-136			URL	URL_2		In this article, we give explicit formulas of archimedean Whittaker functions on $GL(3)$ and $GL(2)$. Moreover, we apply those to the calculation of archimedean zeta integrals for $GL(3)\times GL(2)$, and show that the zeta integral for appropriate Whittaker functions is equal to the associated $L$-factors.
2	RAMANUJAN CAYLEY GRAPHS OF FROBENIUS GROUPS	2016/12	Miki Hirano Kohei Katata Yoshinori Yamasaki	BULLETIN OF THE AUSTRALIAN MATHEMATICAL SOCIETY	94/ 3, 373-383		10.1017/S0004972716000587	URL	URL_2	CAMBRIDGE UNIV PRESS	We determine a bound for the valency in a family of dihedrants of twice odd prime orders which guarantees that the Cayley graphs are Ramanujan graphs. We take two families of Cayley graphs with the underlying dihedral group of order 2p: one is the family of all Cayley graphs and the other is the family of normal ones. In the normal case, which is easier, we discuss the problem for a wider class of groups, the Frobenius groups. The result for the family of all Cayley graphs is similar to that for circulants: the prime p is 'exceptional' if and only if it is represented by one of six specific quadratic polynomials.
3	The archimedean zeta integrals for GL(3) x GL(2)	2016/02	Miki Hirano Taku Ishii Tadashi Miyazaki	PROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES	92/ 2, 27-32	Research paper (scientific journal)	10.3792/pjaa.92.27	URL		JAPAN ACAD	We consider here the archimedean zeta integrals for GL(3) x GL(2) and show that the zeta integral for appropriate Whittaker functions is equal to the associated L-factor.
4	Ramanujan circulant graphs and the conjecture of Hardy-Littlewood and Bateman-Horn	2016	Miki Hirano Kohei Katata Yoshinori Yamasaki	Preprint	arXiv:1310.2130	(MISC) Institution technical report and pre-print, etc.		URL			In this paper, we determine the bound of the valency of the odd circulant graph which guarantees to be a Ramanujan graph for each fixed number of vertices. In almost of the cases, the bound coincides with the trivial bound, which comes from the trivial estimate of the largest non-trivial eigenvalue of the circulant graph. As exceptional cases, the bound in fact exceeds the trivial one by two. We then prove that such exceptionals occur only in the cases where the number of vertices has at most two prime factors and is represented by a quadratic polynomial in a finite family and, moreover, under the conjecture of Hardy-Littlewood and Bateman-Horn, exist infinitely many.
5	無限素点における $GL(3)\times GL(2)$ に関する局所ゼータ積分 (モジュラー形式と保型表現)	2015/11	平野幹石井卓宮崎直	数理解析研究所講究録	1973					京都大学
6	The archimedean Whittaker functions on GL(3)	2012	Miki Hirano Taku Ishii Tadashi Miyazaki	GEOMETRY AND ANALYSIS OF AUTOMORPHIC FORMS OF SEVERAL VARIABLES	7, 77-109	Research paper (international conference proceedings)				WORLD SCIENTIFIC PUBL CO PTE LTD	We introduce the explicit formulas of archimedean Whittaker functions on GL(3) and their application to archimedean zeta integrals.
7	Jackson q-Mahler measures	2010	Miki Hirano Nobushige Kurokawa	Functiones et Approximatio, Commentarii Mathematici	42/ 1, 51-58	Research paper (scientific journal)	10.7169/facm/1269437068			Adam Mickiewicz University Press	In this note, we define a q-analogue of the Mahler measures by using the Jackson integral which we call the Jackson q-Mahler measures. Especially we study their classical limit for polynomials of one variable.
8	Calculus of principal series Whittaker functions on GL(3, C)	2009/04	Miki Hirano Takayuki Oda	JOURNAL OF FUNCTIONAL ANALYSIS	256/ 7, 2222-2267	Research paper (scientific journal)	10.1016/j.jfa.2008.10.011	URL		ACADEMIC PRESS INC ELSEVIER SCIENCE	In this paper, we discuss the Whittaker functions for the non-spherical principal series representations of GL(3, C). In particular, we give explicit formulas for these functions. (c) 2008 Elsevier Inc. All rights reserved.
9	PRINCIPAL SERIES WHITTAKER FUNCTIONS ON $GL$(3, C) (Automorphic Representations, Automorphic Forms, L-functions, and Related Topics)	2008/10	HIRANO MIKI ODA TAKAYUKI	RIMS Kokyuroku	1617					Kyoto University
10	PROPAGATION FORMULA FOR PRINCIPAL SERIES WHITTAKER FUNCTIONS ON GL(3,C)	2007/12	Hirano Miki	Journal of the Faculty of Science and Technology Seikei University	44/ 2			URL	URL_2	Seikei University
11	Whittaker functions for P-j-principal series representations of Sp(3, R)	2007/11	Miki Hirano Taku Ishii Takayuki Oda	ADVANCES IN MATHEMATICS	215/ 2, 734-765	Research paper (scientific journal)	10.1016/j.aim.2007.04.015	URL		ACADEMIC PRESS INC ELSEVIER SCIENCE	In this paper, we give explicit formulas for the secondary and the primary Whittaker functions for P-J-principal series representations of Sp(3, R). (c) 2007 Elsevier Inc. All fights reserved.
12	Confluence from Siegel-Whittaker functions to Whittaker functions on Sp(2, R)	2006/07	Miki Hirano Taku Ishii Takayuki Oda	MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY	141, 15-31	Research paper (scientific journal)	10.1017/S0305004106009224	URL		CAMBRIDGE UNIV PRESS	We discuss a confluence from Siegel-Whittaker functions to Whittaker functions on Sp(2, R) by using their explicit formulae. In our proof, we use expansion theorems of the good Whittaker functions by the secondary Whittaker functions.
13	Secondary Whittaker functions for P-J-principal series representations of Sp(3, R)	2005/06	M Hirano T Oda	PROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES	81/ 6, 105-109	Research paper (scientific journal)	10.3792/pjaa.81.105	URL		JAPAN ACAD	In this paper, we give explicit formulas for the secondary Whittaker functions for P-J-principal series representations of Sp(3, R), which are power series solutions of a holonomic system of rank 24.
14	Confluence from Siegel-Whittaker functions to Whittaker functions on $Sp$(2, $\mathbf{R}$) (Automorphic forms on $Sp$(2,$\mathbf{R}$) and $SU$(2,2) III)	2005/04	Hirano Miki Ishii Taku Oda Takayuki	RIMS Kokyuroku	1421					Kyoto University
15	WHITTAKER FUNCTIONS FOR $P_J$-PRINCIPAL SERIES REPRESENTATIONS OF $Sp$(3, $\mathbf{R}$) (Automorphic forms on $Sp$(2,$\mathbf{R}$) and $SU$(2,2) III)	2005/04	Hirano Miki Oda Takayuki	RIMS Kokyuroku	1421					Kyoto University
16	Fourier-Jacobi type spherical functions for principal series representations of Sp(2, R)	2004	Miki Hirano	Indagationes Mathematicae	15/ 1, 43-54	Research paper (scientific journal)	10.1016/S0019-3577(04)90004-3	URL			In this paper, we study the Fourier-Jacobi type spherical functions on Sp (2, R) for irreducible principal series representations. We give the multiplicity theorem and an explicit formula for this function.
17	Half zeta functions	2003	HIRANO Miki KUROKAWA Nobushige WAKAYAMA Masato	J. Ramanujan Math. Soc.	18/ 2, 195-209
18	Fourier-Jacobi type spherical, functions for P-j-principal series representations of Sp(2,R)	2002/06	M Hirano	JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES	65, 524-546	Research paper (scientific journal)	10.1112/S0024610701002927	URL		LONDON MATH SOC	The paper studies a generalized spherical function, or a generalized Whittaker model for generalized principal series representations of G = Sp(2, R) induced from the Jacobi maximal parabolic subgroup P-J, which is called the Fourier-Jacobi type. In particular, a multiplicity theorem and an explicit formula via the Meijer G-functions for this function are given.
19	Fourier-Jacobi type spherical functions for discrete series representations of Sp(2, R)	2001	M Hirano	COMPOSITIO MATHEMATICA	128/ 2, 177-216	Research paper (scientific journal)	10.1023/A:1017528120756	URL		KLUWER ACADEMIC PUBL	In this paper we define a kind of generalized spherical functions on Sp(2, R). We call it 'Fourier-Jacobi type', since it can be considered as a generalized Whittaker model associated with the Jacobi maximal parabolic subgroup. Also we give the multiplicity theorem and an explicit formula of these functions for discrete series representations of Sp(2, R).
20	Shintani functions on GL(2, C)	2001	M Hirano	TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY	353/ 4, 1535-1550	Research paper (scientific journal)		URL		AMER MATHEMATICAL SOC	In this paper, in analogy to the real case, we give a formulation of the Shintani functions on GL(2, C), which have been studied by Murase and Sugano within the theory of automorphic L-functions. Also, we obtain the multiplicity one theorem for these functions and an explicit formula in a special case.
21	Shintani functions on GL(2; R)	2000	M Hirano	TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY	352/ 4, 1709-1721	Research paper (scientific journal)				AMER MATHEMATICAL SOC	In this paper, we give a formulation and an explicit formula for Shintani function on GL(2, R), which has been studied by Murase and Sugano in the theory of automorphic L-functions. In particular, we obtain the multiplicity of this function.
22	Archimedean Shintani functions on $GL(2)$ (Automorphic Forms and $L$-Functions)	1999/06	Hirano Miki	RIMS Kokyuroku	1103					Kyoto University
23	$GL$(2,$\mathbf{C}$)上の新谷関数 (Sp(2;$\mathbb{R}$)とSU(2,2)上の保型形式 II)	1999/04	平野幹	数理解析研究所講究録	1094					京都大学
24	FOURIER-JACOBI TYPE SPHERICAL FUNCTIONS ON $S_p(2,\mathbf{R})$ ; THE CASE OF $P_J$-PRINCIPAL SERIES AND DISCRETE SERIES (Automorphic Forms and Number Theory)	1998/06	Hirano Miki	RIMS Kokyuroku	1052					Kyoto University
25	On theta type functions associated with the zeros of the Selberg zeta functions	1997/01	M Hirano	MANUSCRIPTA MATHEMATICA	92/ 1, 87-105	Research paper (scientific journal)				SPRINGER VERLAG	In this paper, we consider a kind of theta type function concerning the zeros of the Selberg zeta function. This is obtained from an application of Cartier-Voros type Selberg trace formula for non co-compact but co-finite volume discrete subgroups of PSL(2, R).
26	ON CARTIER-VOROS TYPE SELBERG TRACE FORMULA FOR CONGRUENCE SUBGROUPS OF PSL(2,R)	1995/09	M HIRANO	PROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES	71/ 7, 144-147	Research paper (scientific journal)				JAPAN ACAD

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