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Multiple Gamma Function and Its Application to Computation of Series, preprint
, 2003
"... Abstract. The multiple gamma function Γn, defined by a recurrencefunctional equation as a generalization of the Euler gamma function, was originally introduced by Kinkelin, Glaisher, and Barnes around 1900. Today, due to the pioneer work of Conrey, Katz and Sarnak, interest in the multiple gamma fu ..."
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Abstract. The multiple gamma function Γn, defined by a recurrencefunctional equation as a generalization of the Euler gamma function, was originally introduced by Kinkelin, Glaisher, and Barnes around 1900. Today, due to the pioneer work of Conrey, Katz and Sarnak, interest in the multiple gamma function has been revived. This paper discusses some theoretical aspects of the Γn function and their applications to summation of series and infinite products.
Maxwell strata in subRiemannian problem on the group of motions of a plane
, 2008
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Conjugate points in Euler’s elastic problem
 Journal of Dynamical and Control Systems (accepted), available at: arXiv:0705.1003
"... Abstract. For the classical Euler elastic problem, conjugate points are described. Inflexional elasticas admit the first conjugate point between the first and third inflexion points. All other elasticas do not have conjugate points. As a result, the problem of stability of Euler elasticas is solved. ..."
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Abstract. For the classical Euler elastic problem, conjugate points are described. Inflexional elasticas admit the first conjugate point between the first and third inflexion points. All other elasticas do not have conjugate points. As a result, the problem of stability of Euler elasticas is solved. 1.
The Wilson function transform
 Int. Math. Res. Not. 2003
"... Abstract. Two unitary integral transforms with a verywell poised 7F6function as a kernel are given. For both integral transforms the inverse is the same as the original transform after an involution on the parameters. The 7F6function involved can be considered as a nonpolynomial extension of the ..."
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Cited by 12 (2 self)
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Abstract. Two unitary integral transforms with a verywell poised 7F6function as a kernel are given. For both integral transforms the inverse is the same as the original transform after an involution on the parameters. The 7F6function involved can be considered as a nonpolynomial extension of the Wilson polynomial, and is therefore called a Wilson function. The two integral transforms are called a Wilson function transform of type I and type II. Furthermore, a few explicit transformations of hypergeometric functions are calculated, and it is shown that the Wilson function transform of type I maps a basis of orthogonal polynomials onto a similar basis of polynomials. 1.
Measures of Distinctness for Random Partitions and Compositions of an Integer
, 1997
"... This paper is concerned with problems of the following type: # Accepted for publication in Advances in Applied Mathematics. Given a random (under a suitable probability model) partition or composition, study quantitatively the measures of the degree of distinctness of its parts ..."
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Cited by 12 (2 self)
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This paper is concerned with problems of the following type: # Accepted for publication in Advances in Applied Mathematics. Given a random (under a suitable probability model) partition or composition, study quantitatively the measures of the degree of distinctness of its parts
Root asymptotics of spectral polynomials for the Lamé operator
 Commun. Math. Phys
"... Abstract. The study of polynomial solutions to the classical Lamé equation in its algebraic form, or equivalently, of doubleperiodic solutions of its Weierstrass form has a long history. Such solutions appear at integer values of the spectral parameter and their respective eigenvalues serve as the ..."
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Abstract. The study of polynomial solutions to the classical Lamé equation in its algebraic form, or equivalently, of doubleperiodic solutions of its Weierstrass form has a long history. Such solutions appear at integer values of the spectral parameter and their respective eigenvalues serve as the ends of bands in the boundary value problem for the corresponding Schrödinger equation with finite gap potential given by the Weierstrass ℘function on the real line. In this paper we establish several natural (and equivalent) formulas in terms of hypergeometric and elliptic type integrals for the density of the appropriately scaled asymptotic distribution of these eigenvalues when the integervalued spectral parameter tends to infinity. We also show that this density satisfies a Heun differential equation with four singularities.
APPLICATIONS OF DIFFERENTIAL SUBORDINATION TO CERTAIN SUBCLASSES OF MEROMORPHICALLY MULTIVALENT FUNCTIONS
"... ABSTRACT. By making use of the principle of differential subordination, the authors investigate several inclusion relationships and other interesting properties of certain subclasses of meromorphically multivalent functions which are defined here by means of a linear operator. They also indicate rel ..."
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Cited by 11 (1 self)
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ABSTRACT. By making use of the principle of differential subordination, the authors investigate several inclusion relationships and other interesting properties of certain subclasses of meromorphically multivalent functions which are defined here by means of a linear operator. They also indicate relevant connections of the various results presented in this paper with those obtained in earlier works.
New blowup rates for fast controls of Schrödinger and heat equations, Les prépublications de l’Institut Elie
 Cartan
, 2007
"... Abstract: We consider the nullcontrollability problem for the Schrödinger and heat equations with boundary control. We concentrate on shorttime, or fast, controls. We improve recent estimates (see Miller [14], [15],[16] [17]) on the norm of the operator associating to any initial state the minimal ..."
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Abstract: We consider the nullcontrollability problem for the Schrödinger and heat equations with boundary control. We concentrate on shorttime, or fast, controls. We improve recent estimates (see Miller [14], [15],[16] [17]) on the norm of the operator associating to any initial state the minimal norm control driving the system to zero. Our main results concern the Schrödinger and heat equations in one space dimension. They yield new estimates concerning window problems for series of exponentials as described in Seidman, Avdonin and Ivanov [22]. These results are used, following [17], to deal with the case of several space dimensions.
The height of watermelons with wall
"... Abstract. We derive asymptotics for the moments as well as the weak limit of the height distribution of watermelons with p branches with wall. This generalises a famous result of de Bruijn, Knuth and Rice [4] on the average height of planted plane trees, and results by Fulmek [9] and Katori et al. [ ..."
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Abstract. We derive asymptotics for the moments as well as the weak limit of the height distribution of watermelons with p branches with wall. This generalises a famous result of de Bruijn, Knuth and Rice [4] on the average height of planted plane trees, and results by Fulmek [9] and Katori et al. [14] on the expected value, respectively higher moments, of the height distribution of watermelons with two branches. The asymptotics for the moments depend on the analytic behaviour of certain multidimensional Dirichlet series. In order to obtain this information we prove a reciprocity relation satisfied by the derivatives of one of Jacobi’s theta functions, which generalises the well known reciprocity law for Jacobi’s theta functions. 1.
Contribution to the Theory of the Barnes Function
"... Abstract. This paper presents a family of new integral representations and asymptotic series of the multiple gamma function. The numerical schemes for highprecision computation of the Barnes gamma function and Glaisher’s constant are also discussed. 1. ..."
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Abstract. This paper presents a family of new integral representations and asymptotic series of the multiple gamma function. The numerical schemes for highprecision computation of the Barnes gamma function and Glaisher’s constant are also discussed. 1.