# The leading term of the Plancherel-Rotach asymptotic formula for solutions of recurrence relations

## Abstract

Recurrence relations generating Padé and Hermite-Padé polynomials are considered. Their coefficients increase with the index of the relation, but after dividing by an appropriate power of the scaling function they tend to a finite limit. As a result, after scaling the polynomials 'stabilize' for large indices; this type of asymptotic behaviour is called Plancherel-Rotach asymptotics. An explicit expression for the leading term of the asymptotic formula, which is valid outside sets containing the zeros of the polynomials, is obtained for wide classes of three- and four-term relations. For three-term recurrence relations this result generalizes a theorem Van Assche obtained for recurrence relations with 'regularly' growing coefficients. Bibliography: 19 titles.

- Authors:

- M.V. Keldysh Institute for Applied Mathematics, Russian Academy of Sciences, Moscow (Russian Federation)

- Publication Date:

- OSTI Identifier:
- 22364154

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Sbornik. Mathematics; Journal Volume: 205; Journal Issue: 12; Other Information: Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICAL METHODS AND COMPUTING; ASYMPTOTIC SOLUTIONS; INDEXES; PADE APPROXIMATION; POLYNOMIALS; RECURSION RELATIONS

### Citation Formats

```
Aptekarev, A I, and Tulyakov, D N.
```*The leading term of the Plancherel-Rotach asymptotic formula for solutions of recurrence relations*. United States: N. p., 2014.
Web. doi:10.1070/SM2014V205N12ABEH004435.

```
Aptekarev, A I, & Tulyakov, D N.
```*The leading term of the Plancherel-Rotach asymptotic formula for solutions of recurrence relations*. United States. doi:10.1070/SM2014V205N12ABEH004435.

```
Aptekarev, A I, and Tulyakov, D N. Wed .
"The leading term of the Plancherel-Rotach asymptotic formula for solutions of recurrence relations". United States.
doi:10.1070/SM2014V205N12ABEH004435.
```

```
@article{osti_22364154,
```

title = {The leading term of the Plancherel-Rotach asymptotic formula for solutions of recurrence relations},

author = {Aptekarev, A I and Tulyakov, D N},

abstractNote = {Recurrence relations generating Padé and Hermite-Padé polynomials are considered. Their coefficients increase with the index of the relation, but after dividing by an appropriate power of the scaling function they tend to a finite limit. As a result, after scaling the polynomials 'stabilize' for large indices; this type of asymptotic behaviour is called Plancherel-Rotach asymptotics. An explicit expression for the leading term of the asymptotic formula, which is valid outside sets containing the zeros of the polynomials, is obtained for wide classes of three- and four-term relations. For three-term recurrence relations this result generalizes a theorem Van Assche obtained for recurrence relations with 'regularly' growing coefficients. Bibliography: 19 titles.},

doi = {10.1070/SM2014V205N12ABEH004435},

journal = {Sbornik. Mathematics},

number = 12,

volume = 205,

place = {United States},

year = {Wed Dec 31 00:00:00 EST 2014},

month = {Wed Dec 31 00:00:00 EST 2014}

}