28,789 research outputs found
Propagation of generalized vector Helmholtz-Gauss beams through paraxial optical systems
We introduce the generalized vector Helmholtz-Gauss (gVHzG) beams that constitute a general family of localized beam solutions of the Maxwell equations in the paraxial domain. The propagation of the electromagnetic components through axisymmetric ABCD optical systems is expressed elegantly in a coordinate-free and closed-form expression that is fully characterized by the transformation of two independent complex beam parameters. The transverse mathematical structure of the gVHzG beams is form-invariant under paraxial transformations. Any paraxial beam with the same waist size and transverse spatial frequency can be expressed as a superposition of gVHzG beams with the appropriate weight factors. This formalism can be straightforwardly applied to propagate vector Bessel-Gauss, Mathieu-Gauss, and Parabolic-Gauss beams, among others
Special function identities from superelliptic Kummer varieties
We prove that the factorization of Appell's generalized hypergeometric series
satisfying the so-called quadric property into a product of two Gauss'
hypergeometric functions has a geometric origin: we first construct a
generalized Kummer variety as minimal nonsingular model for a product-quotient
surface with only rational double points from a pair of superelliptic curves of
genus with . We then show that this generalized Kummer
variety is equipped with two fibrations with fibers of genus . When
periods of a holomorphic two-form over carefully crafted transcendental
two-cycles on the generalized Kummer variety are evaluated using either of the
two fibrations, the answer must be independent of the fibration and the
aforementioned family of special function identities is obtained. This family
of identities can be seen as a multivariate generalization of Clausen's
Formula. Interestingly, this paper's finding bridges Ernst Kummer's two
independent lines of research, algebraic transformations for the Gauss'
hypergeometric function and nodal surfaces of degree four in .Comment: 46 pages, 2 figure
Clausen's series 3F2(1) with integral parameter differences and transformations of the hypergeometric function 2F2(x)
We obtain summation formulas for the hypergeometric series 3 F 2(1) with at least one pair of numeratorial and denominatorial parameters differing by a negative integer. The results derived for the latter are used to obtain Kummer-type transformations for the generalized hypergeometric function 2 F 2(x) and reduction formulas for certain Kampé de Fériet functions. Certain summations for the partial sums of the Gauss hypergeometric series 2 F 1(1) are also obtained
Some Quadratic Transformations and Reduction Formulas associated with Hypergeometric Functions
In this paper, we construct four summation formulas for terminating Gaussâ hypergeometric series having argument âtwo and with the help of our summation formulas. We establish two quadratic transformations for Gaussâ hypergeometric function in terms of finite summation of combination of two Clausen hypergeometric functions. Further, we have generalized our quadratic transformations in terms of general double series identities as well as in terms of reduction formulas for KampĂ© de FĂ©rietâs double hypergeometric function. Some results of Rathie-Nagar, Kim et al. and Choi-Rathie are also obtained as special cases of our findings
Airy-Gauss beams and their transformation by paraxial optical systems
We introduce the generalized Airy-Gauss (AiG) beams and analyze their propagation through optical systems described by ABCD matrices with complex elements in general. The transverse mathematical structure of the AiG beams is form-invariant under paraxial transformations. The conditions for square integrability of the beams are studied in detail. The model of the AiG beam describes in a more realistic way the propagation of the Airy wave packets because AiG beams carry finite power, retain the nondiffracting propagation properties within a finite propagation distance, and can be realized experimentally to a very good approximation
Multiple orthogonal polynomials of mixed type: Gauss-Borel factorization and the multi-component 2D Toda hierarchy
Multiple orthogonality is considered in the realm of a Gauss--Borel
factorization problem for a semi-infinite moment matrix. Perfect combinations
of weights and a finite Borel measure are constructed in terms of M-Nikishin
systems. These perfect combinations ensure that the problem of mixed multiple
orthogonality has a unique solution, that can be obtained from the solution of
a Gauss--Borel factorization problem for a semi-infinite matrix, which plays
the role of a moment matrix. This leads to sequences of multiple orthogonal
polynomials, their duals and second kind functions. It also gives the
corresponding linear forms that are bi-orthogonal to the dual linear forms.
Expressions for these objects in terms of determinants from the moment matrix
are given, recursion relations are found, which imply a multi-diagonal Jacobi
type matrix with snake shape, and results like the ABC theorem or the
Christoffel--Darboux formula are re-derived in this context (using the
factorization problem and the generalized Hankel symmetry of the moment
matrix). The connection between this description of multiple orthogonality and
the multi-component 2D Toda hierarchy, which can be also understood and studied
through a Gauss--Borel factorization problem, is discussed. Deformations of the
weights, natural for M-Nikishin systems, are considered and the correspondence
with solutions to the integrable hierarchy, represented as a collection of Lax
equations, is explored. Corresponding Lax and Zakharov--Shabat matrices as well
as wave functions and their adjoints are determined. The construction of
discrete flows is discussed in terms of Miwa transformations which involve
Darboux transformations for the multiple orthogonality conditions. The bilinear
equations are derived and the -function representation of the multiple
orthogonality is given.Comment: 53 pages. In this version minor revisions regarding the
Christoffel-Darboux operators are performe
- âŠ