Bi-axial Gegenbauer functions of the first and second kind

Alan Common

Banach Center Publications (1996)

  • Volume: 37, Issue: 1, page 181-187
  • ISSN: 0137-6934

Abstract

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The classical orthogonal polynomials defined on intervals of the real line are related to many important branches of analysis and applied mathematics. Here a method is described to generalise this concept to polynomials defined on higher dimensional spaces using Bi-Axial Monogenic functions. The particular examples considered are Gegenbauer polynomials defined on the interval [-1,1] and the Gegenbauer functions of the second kind which are weighted Cauchy integral transforms over this interval of these polynomials. Related polynomials are defined which are orthogonal on the unit ball p ≡ x → ∈ ℝ p ; | x → | ≤ 1 using Bi-Axial Monogenic generating functions on ℝ m . Then corresponding generalised Gegenbauer functions of the second kind are defined using generalised weighted Bi-Axial Monogenic Cauchy transforms of these polynomials over p . These generalised Gegenbauer functions of first and second kind reduce to the standard case when p=1 and are solutions of related second order differential equations which become identical in the one dimensional case.

How to cite

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Common, Alan. "Bi-axial Gegenbauer functions of the first and second kind." Banach Center Publications 37.1 (1996): 181-187. <http://eudml.org/doc/208595>.

@article{Common1996,
abstract = {The classical orthogonal polynomials defined on intervals of the real line are related to many important branches of analysis and applied mathematics. Here a method is described to generalise this concept to polynomials defined on higher dimensional spaces using Bi-Axial Monogenic functions. The particular examples considered are Gegenbauer polynomials defined on the interval [-1,1] and the Gegenbauer functions of the second kind which are weighted Cauchy integral transforms over this interval of these polynomials. Related polynomials are defined which are orthogonal on the unit ball $^\{p\} ≡ \{\vec\{x\} ∈ ℝ^\{p\} ; |\vec\{x\}| ≤ 1\}$ using Bi-Axial Monogenic generating functions on $ℝ^\{m\}$. Then corresponding generalised Gegenbauer functions of the second kind are defined using generalised weighted Bi-Axial Monogenic Cauchy transforms of these polynomials over $^\{p\}$. These generalised Gegenbauer functions of first and second kind reduce to the standard case when p=1 and are solutions of related second order differential equations which become identical in the one dimensional case.},
author = {Common, Alan},
journal = {Banach Center Publications},
keywords = {Gegenbauer polynomials},
language = {eng},
number = {1},
pages = {181-187},
title = {Bi-axial Gegenbauer functions of the first and second kind},
url = {http://eudml.org/doc/208595},
volume = {37},
year = {1996},
}

TY - JOUR
AU - Common, Alan
TI - Bi-axial Gegenbauer functions of the first and second kind
JO - Banach Center Publications
PY - 1996
VL - 37
IS - 1
SP - 181
EP - 187
AB - The classical orthogonal polynomials defined on intervals of the real line are related to many important branches of analysis and applied mathematics. Here a method is described to generalise this concept to polynomials defined on higher dimensional spaces using Bi-Axial Monogenic functions. The particular examples considered are Gegenbauer polynomials defined on the interval [-1,1] and the Gegenbauer functions of the second kind which are weighted Cauchy integral transforms over this interval of these polynomials. Related polynomials are defined which are orthogonal on the unit ball $^{p} ≡ {\vec{x} ∈ ℝ^{p} ; |\vec{x}| ≤ 1}$ using Bi-Axial Monogenic generating functions on $ℝ^{m}$. Then corresponding generalised Gegenbauer functions of the second kind are defined using generalised weighted Bi-Axial Monogenic Cauchy transforms of these polynomials over $^{p}$. These generalised Gegenbauer functions of first and second kind reduce to the standard case when p=1 and are solutions of related second order differential equations which become identical in the one dimensional case.
LA - eng
KW - Gegenbauer polynomials
UR - http://eudml.org/doc/208595
ER -

References

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  1. [1] A. K. Common and F. Sommen, Axial Monogenic Functions from Holomorphic Functions, J. Math. Anal. Appl. 179 (1993), 610-629. Zbl0802.30001
  2. [2] A. K. Common and F. Sommen, Special Bi-Axial Monogenic Functions, J. Math. Anal. Appl. 185 (1994), 189-206. 
  3. [3] A. K. Common and F. Sommen, Cauchy transforms and bi-axial monogenic power functions, Proccedings of Dienze conference on Clifford Algebras and Their Applications in Mathematical Physics, Kluwer, 1993, 85-90. Zbl0840.30029
  4. [4] A. K. Common and F. Sommen, Bi-axial Gegenbauer functions of the second kind, to be published in: Journal of Mathematical Analysis and Applications. 
  5. [5] J. Cnops, Orthogonal polynomials associated with the dirac operator in euclidean space, Chinese Ann.Math. 13B:1 (1992), 68-79. 
  6. [6] H. Hochstadt, The Functions of Mathematical Physics, (1971) Wiley-Interscience, New York. Zbl0217.39501
  7. [7] G. Jank and F. Sommen, Clifford analysis, bi-axial symmetry and pseudo-analytic functions, Complex Variables 13 (1990), 195-212. Zbl0703.30044
  8. [8] F. Sommen, Plane elliptic systems and monogenic functions in symmetric domains, Supp. Rend. Circ. Mat. Palermo. 6 (1984), 259-269. Zbl0564.30036

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