The effective formulas reducing the two‐dimensional Hermite polynomials to the classical one‐dimensional orthogonal polynomials by Jacobi, Gegenbauer, Legendre, Laguerre, and Hermite are given. New one‐parameter generating functions for the ‘‘diagonal’’ multidimensional Hermite polynomials are derived. The factorial moments and cumulants of the distribution functions related to the Hermite polynomials of two variables with equal indices are expressed in terms of the Legendre and Chebyshev polynomials. Asymptotical formulas for the two‐dimensional polynomials with large values of indices and zero arguments are found. The applications to the squeezed one‐mode states and to the time‐dependent quantum harmonic oscillator are considered.

1.
I. A.
Malkin
,
V. I.
Man’ko
, and
D. A.
Trifonov
,
J. Math. Phys.
14
,
576
(
1973
).
2.
I. A. Malkin and V. I. Man’ko, Dynamical Symmetries and Coherent States of Quantum Systems (Nauka, Moscow, 1979) (in Russian).
3.
V. V. Dodonov and V. I. Man’ko, in Invariants and Evolution of Nonstationary Quantum Systems, Proceedings of Lebedev Physics Institute, edited by M. A. Markov (Nova Science, Commack, NY, 1989), Vol. 183, p. 3 (English Translation).
4.
M.
Kauderer
,
J. Math. Phys.
34
,
4221
(
1993
).
5.
A.
Vourdas
and
R. M.
Weiner
,
Phys. Rev.
36
,
5866
(
1987
).
6.
G.
Adam
,
Phys. Lett. A
171
,
66
(
1992
).
7.
I. A.
Malkin
,
V. I.
Man’ko
, and
D. A.
Trifonov
,
Phys. Rev. D
2
,
1371
(
1970
).
8.
V. V.
Dodonov
,
I. A.
Malkin
, and
V. I.
Man’ko
,
Physica
59
,
241
(
1972
).
9.
V. V.
Dodonov
,
V. I.
Man’ko
, and
V. V.
Semjonov
,
Nuovo Cimento B
83
,
145
(
1984
).
10.
V. V. Dodonov, V. I. Man’ko, and I. N. Prokopenya, Asymptotic Density Matrix Elements of the Canonically Transformed Oscillator Hamiltonian in the Fock Basis, Preprint No. 30, Lebedev Physics Institute, Moscow, 1985.
11.
A.
Vourdas
,
Phys. Rev. A
34
,
3466
(
1986
).
12.
Bateman Manuscript Project: Higher Transcendental Functions, edited by A. Erdélyi (McGraw-Hill, New York, 1953).
13.
G. Szegö, Orthogonal Polynomials (American Mathematical Society, New York, 1959).
14.
P. Appel and J. Kampé de Fériet, Fonctions Hypergéométriques et Hypersphériques. Polynomes d’Hermite (Gauthier-Villars, Paris, 1926).
15.
F. W. J. Olver, Asymptotics and Special Functions (Academic, New York, 1974).
16.
Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun (National Bureau of Standards, Washington, D. C., 1964).
17.
S.
Chaturvedi
and
V.
Srinivasan
,
Phys. Rev. A
40
,
6095
(
1989
).
18.
V. V.
Dodonov
,
O. V.
Man’ko
, and
V. I.
Man’ko
,
Phys. Rev. A
49
,
2993
(
1994
).
19.
P.
Marian
and
T. A.
Marian
,
Phys. Rev. A
47
,
4474
(
1993
).
20.
W.
Schleich
and
J. A.
Wheeler
,
J. Opt. Soc. Am. B
4
,
1715
(
1987
).
21.
V. V.
Dodonov
,
A. B.
Klimov
, and
V. I.
Man’ko
,
Phys. Lett. A
134
,
211
(
1989
).
22.
V. V. Dodonov, A. B. Klimov, and V. I. Man’ko, in Squeezed and Correlated States of Quantum Systems, Proceedings of Lebedev Physics Institute, edited by M. A. Markov (Nova Science, Commack, 1993), Vol. 205, p. 61.
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