Generalized integrals of Macdonald and Gegenbauer functions
Publikation: Working paper › Preprint › Forskning
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Generalized integrals of Macdonald and Gegenbauer functions. / Dereziński, Jan; Gaß, Christian; Ruba, Błażej.
arXiv.org, 2023.Publikation: Working paper › Preprint › Forskning
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TY - UNPB
T1 - Generalized integrals of Macdonald and Gegenbauer functions
AU - Dereziński, Jan
AU - Gaß, Christian
AU - Ruba, Błażej
N1 - 39 pages
PY - 2023/4/12
Y1 - 2023/4/12
N2 - We compute bilinear integrals involving Macdonald and Gegenbauer functions. These integrals are convergent only for a limited range of parameters. However, when one uses generalized integrals they can be computed essentially without restricting the parameters. The generalized integral is a linear functional extending the standard integral to a certain class of functions involving finitely many homogeneous non-integrable terms at the edpoints of the interval. For generic values of parameters, generalized bilinear integrals of Macdonald and Gegenbauer functions can be obtained by analytic continuation from the region in which the integrals are convergent. In the case of integer parameters we obtain expressions with explicit additional terms related to an anomaly, namely the failure of the generalized integral to be scaling invariant.
AB - We compute bilinear integrals involving Macdonald and Gegenbauer functions. These integrals are convergent only for a limited range of parameters. However, when one uses generalized integrals they can be computed essentially without restricting the parameters. The generalized integral is a linear functional extending the standard integral to a certain class of functions involving finitely many homogeneous non-integrable terms at the edpoints of the interval. For generic values of parameters, generalized bilinear integrals of Macdonald and Gegenbauer functions can be obtained by analytic continuation from the region in which the integrals are convergent. In the case of integer parameters we obtain expressions with explicit additional terms related to an anomaly, namely the failure of the generalized integral to be scaling invariant.
KW - math.CA
KW - math-ph
KW - math.MP
KW - 33C05, 33C10, 47A52
M3 - Preprint
BT - Generalized integrals of Macdonald and Gegenbauer functions
PB - arXiv.org
ER -
ID: 382552940