## Effective Laguerre asymptotics (2007)

### Cached

### Download Links

Citations: | 1 - 1 self |

### BibTeX

@MISC{Borwein07effectivelaguerre,

author = {D. Borwein and Jonathan M. Borwein and Richard E. Crandall},

title = {Effective Laguerre asymptotics},

year = {2007}

}

### OpenURL

### Abstract

It is known that the generalized Laguerre polynomials can enjoy sub-exponential growth for large primary index. Specifically, for certain fixed parameter pairs (a, z) one has the large-n asymptotic L (−a) n (−z) ∼ C(a, z)n −a/2−1/4 e 2√nz. We introduce a computationally motivated contour integral that allows highly efficient numerical evaluation of Ln, yet also leads to general asymptotic series over the full domain for sub-exponential behavior. We eventually lay out a fast algorithm for generation of the rather formidable expansion coefficients. Along the way we address the difficult problem of establishing effective (i.e. rigorous and explicit) error bounds on the general expansion. To this end, we avoid classical stationary-phase and steepest-descent techniques in favor of an “exp-arc ” method that amounts to a natural bridge between converging series and effective asymptotics. Finally, we exhibit an absolutely convergent exp-arc series for Bessel-function evaluation as an alternative to conventional ascending-asymptotic switching.

### Citations

447 |
Asymptotics and special functions
- Olver
- 1974
(Show Context)
Citation Context ...r our modern purposes), the classical authors certainly knew in principle how to establish effective error bounds. The excellent treatment of effectiveness for Laplace’s method of steepest descent in =-=[22]-=- is a shining example. Even more illuminating is Olver’s paper [21], which explains effective bounding and shows how unwieldy rigorous bounds can be. However, efficient algorithms for generating expli... |

198 |
Applied and computational complex analysis
- Henrici
- 1986
(Show Context)
Citation Context ...valuations [23]. Our own research motive for providing effective asymptotics involves not the Laguerre–Hermite connection, rather it begins with a beautiful link the incomplete gamma function, namely =-=[18]-=- Γ(a, z) = z a e −z 1 = z 1 − a z + 1 1 + 2 − a z + 1 + · · · a e −z ∞� (1 − a)n 1 (n + 1)! n=0 L (−a) n (−z) L (−a) n+1 (−z), (5) where (c)n := c(c + 1) · · · (c + n − 1) is the Pochhammer symbol. Th... |

134 |
Theory of Bessel functions, 2nd edition
- Watson
- 1994
(Show Context)
Citation Context ... to Bessel functions The history of Bessel-function asymptotics is one of the great success stories in the annals of analysis. As early as 1823, Poisson developed the beginnings of Bessel asymptotics =-=[36]-=- [32], and eventually Hankel developed the classic, complete asymptotic series [1, §9.2.5], and effective bounds on error terms (for certain parameter domains) are well known, and in many cases optima... |

91 |
and the AGM: A Study in Analytic Number Theory and Computational Complexity
- Borwein, Borwein, et al.
- 1987
(Show Context)
Citation Context ...the defining series (1) was, in our trials, three-times slower than the integral c1 (the other two contour segments are well below the 13-digit significance). 6 6 Of course, series acceleration as in =-=[3]-=- would give the direct series a “leg up.” Still, it is remarkable that contour integration would even be competitive. For the record, we compared Mathematica’s numerical integration to its own Laguerr... |

65 | Continued Fractions with Applications - Lorentzen, Waadeland - 1992 |

64 |
The Theory of the Riemann Zeta Function
- Titchmarsh
- 1986
(Show Context)
Citation Context ...le, with |F ′′ (x)| ≥ ρ > 0 for x on real interval (a, b). If a real function G(x) on said interval has G/F ′ is monotonic, |G| ≤ M, then � b a G(x)e iF (x) dx = Θ : 8M √ ρ . Proof. This is proved in =-=[33]-=-. QED Then, we follow with an implication specific to the present work: Lemma 16 For any complex p with ℜ(p) ≥ 0, and positive real ν, we have Bν(p) := � √ 1/ 2 x 0 2ν e −2px2 dx = Θ : 9 2ν 1 � . |p| ... |

45 |
Lectures on Hermite and Laguerre expansions
- Thangavelu
- 1993
(Show Context)
Citation Context ...essel functions The history of Bessel-function asymptotics is one of the great success stories in the annals of analysis. As early as 1823, Poisson developed the beginnings of Bessel asymptotics [36] =-=[32]-=-, and eventually Hankel developed the classic, complete asymptotic series [1, §9.2.5], and effective bounds on error terms (for certain parameter domains) are well known, and in many cases optimal [36... |

24 |
Asymptotic estimates of Stirling numbers
- Temme
- 1993
(Show Context)
Citation Context ... number of the first kind, normalized via x(x − 1) · · · (x − h + 1) =: �h j=0 S(j) h xj . So the effective of our lemma is a rigorous bound on the growth of Stirling numbers; see [1, 24.1.3,III] and =-=[31]-=- for research on Stirling asymptotics. Proof. The first Θ-estimate arises by induction. For notational convenience we omit the vector �1 and just use the symbol Gh(j). Note GN(1) = 1/N and GN(2) = N−1... |

22 | Integrals of the Ising class - Bailey, Borwein, et al. |

17 | Riemann–Hilbert analysis for Laguerre polynomials with large negative parameter
- Kuijlaars, McLaughlin
- 2001
(Show Context)
Citation Context ...pplied to such variants. Other important “linked” cases include the exponential monomial and the partial-exponential sum, respectively: L (−n) n (z) = (−1)n n! z n ; L (−n−1) n (z) = (−1) n The paper =-=[16]-=- discusses such generalizations. An important point here is that any asymptotic theory with n ≈ a would have to take into account these useful special cases. On the subject of rigorous bounds, there i... |

10 |
Uniform asymptotic expansions of Laguerre polynomials
- Frenzen, Wong
- 1988
(Show Context)
Citation Context ...on Laguerre asymptotics have emerged from differential theory—there is the classical work of Erdélyi and Olver, plus modern work on combinations of differential, discrete, and saddle-point theory [11]=-=[13]-=-. Next, for some hypergeometric connections, there is a hypergeometric form L (−a) � � n − a n (−z) = 1F1 (−n, 1 − a; −z) , (13) n where 1F1 is also known as the Kummer (confluent) hypergeometric func... |

9 |
Asymptotic forms for Laguerre polynomials
- Erdélyi
- 1960
(Show Context)
Citation Context ...lts on Laguerre asymptotics have emerged from differential theory—there is the classical work of Erdélyi and Olver, plus modern work on combinations of differential, discrete, and saddle-point theory =-=[11]-=-[13]. Next, for some hypergeometric connections, there is a hypergeometric form L (−a) � � n − a n (−z) = 1F1 (−n, 1 − a; −z) , (13) n where 1F1 is also known as the Kummer (confluent) hypergeometric ... |

5 |
On the dynamics of certain recurrence relations
- Borwein, Borwein, et al.
(Show Context)
Citation Context ...nferred from our caveat (a, z) ∈ D. The reason we have emphasized here the rn-iteration (10) itself is that direct analysis of second-order recurrences have, on other problems, yielded strong results =-=[5, 6, 7]-=-. A “discrete” approach that attempts alternative asymptotic expressions for the rn is therefore promising. Perhaps just as promising for growth analysis is the Laguerre differential equation, which i... |

5 |
Strong asymptotics for Laguerre polynomials with varying weights
- Bosbach, Gawronski
- 1998
(Show Context)
Citation Context ...x n is linked either to a or z, or both. For example, the paper [17] provides a large-n asymptotic expansion for L (a) n (nx). Another treatment is [29], in which the author handles L (nx) n (ny). In =-=[8]-=- the authors analyze cases of L (an) n for which an >> n. One hopes that our present methods for effective bounding can be applied to such variants. Other important “linked” cases include the exponent... |

5 |
Strong approximation of eigenvalues of large dimensional Wishart matrices by roots of generalized Laguerre polynomials
- DETTE
- 2002
(Show Context)
Citation Context ...a := 0 the Laguerre zeros are all real and negative; see [20, Ch. X] and references therein. There is also an interesting connection between Laguerre zeros and eigenvalues of certain (large) matrices =-=[10]-=-. • How can the discrete iteration (10) be used directly to glean information about sub-exponential growth? One would think that insertion of a formal asymptotic form into the recursion would force ce... |

5 |
Asymptotic expansions of Charlier, Laguerre and Jacobi polynomials. Submited
- Lopez, Temme
- 2002
(Show Context)
Citation Context ...ve Laguerre studies abound. One modern thrust—which we do not address in the present paper—involves asymptotic behavior when the subindex n is linked either to a or z, or both. For example, the paper =-=[17]-=- provides a large-n asymptotic expansion for L (a) n (nx). Another treatment is [29], in which the author handles L (nx) n (ny). In [8] the authors analyze cases of L (an) n for which an >> n. One hop... |

5 |
Unified algorithms for polylogarithm, L-series, and zeta variants. 2012. Available at http://www.perfscipress.com/papers/UniversalTOC25.pdf
- Crandall
(Show Context)
Citation Context ...to Riemann himself [41, p. 22]. But when s has large imaginary height, one becomes interested in the incomplete gamma’s corresponding complex parameters (a, z) also of possibly large imaginary height =-=[10]-=-[11]. For example, in the art of prime-number counting, say for primes < 10 20 , one might need to know such as Γ � 3/4 + 10 10 i, z � , for various z also of large imaginary height, to good precision... |

5 | Computing the incomplete gamma function to arbitrary precision
- Winitzki
- 2003
(Show Context)
Citation Context ...:= n + 1, the coefficients Ck in the above formula differ slightly from the historical ones. A modern literature treatment that is again consistent with the heuristic (3)–(4), is given by Winitzki in =-=[45]-=-, where one invokes a formal generating function to yield a contour integral for L (−a) n (−z). Then a stationary-phase approach yields the correct first-asymptotic term, at least for certain subregio... |

3 | On the Ramanujan AGM fraction
- Borwein, Crandall
(Show Context)
Citation Context ...nferred from our caveat (a, z) ∈ D. The reason we have emphasized here the rn-iteration (10) itself is that direct analysis of second-order recurrences have, on other problems, yielded strong results =-=[5, 6, 7]-=-. A “discrete” approach that attempts alternative asymptotic expressions for the rn is therefore promising. Perhaps just as promising for growth analysis is the Laguerre differential equation, which i... |

3 |
Explicit bounds for Hermite polynomials in the oscillatory region
- Foster, Krasikov
(Show Context)
Citation Context ...simply asigns x := √ −z with our branch rule √ −ρ := i √ ρ for positive real ρ). One interesting work—whose methods are distinct from our present ones—involves explicit bounds on Hermite oscillations =-=[12]-=-. And, for asymptotic analysis, there may be some hope in the interesting López–Temme expansion: L (−a) n (−z) = (−x) n n� k=0 ckHn−k � a−z−1 2x x k (n − k)! where x := � −z − (1 − a)/2 and the ck are... |

3 |
Approximations of orthogonal polynomials in terms of Hermite polynomials
- Lopez, Temme
- 1999
(Show Context)
Citation Context ...in the interesting López–Temme expansion: L (−a) n (−z) = (−x) n n� k=0 ckHn−k � a−z−1 2x x k (n − k)! where x := � −z − (1 − a)/2 and the ck are generated by a certain 4-th order recurrence relation =-=[18]-=-. Strikingly nonstandard as this representation may be, it is neverthless valid for all parameter pairs (a, z) ∈ C × C. 1.2 Historical results Laguerre asymptotics have long been established for certa... |

3 |
Über das Verhalten einer ausgearteten hypergeometrischen Reihebei unbegrenztem Wachstum eines Parameters
- PERRON
- 1920
(Show Context)
Citation Context ...) where we again use index m := n+1, which slightly alters the coefficients in such classical expansions, said coefficients being for powers n −k/2 but we are now using powers m −k/2 . By 1921 Perron =-=[28]-=- had generalized the Fejér series to arbitrary orders, then for z �∈ (−∞, 0] established a series consistent with (3) and (4), in essentially the form: [30, Theorem 8.22.3] L (−a) n (−z) = Sn(a, z) � ... |

3 | fourth edition - Society, Providence - 1975 |

3 |
A new bound for the smallest x with π(x) > li(x). arXiv:math/0509312 [math.NT], Submitted
- Chao, Plymen
- 2005
(Show Context)
Citation Context ...hat a long formula g can run arbitrarily to the right of the colon. Incidentally there is a literature precedent for such a “theta-calculus.” Namely, in computational number theory treatments such as =-=[10]-=- one encounters terms such as, say, θx/ log 2 x, with a stated constraint θ ∈ [−10, 10]. This means the term is O(x/ log 2 x) but with effective big-O constant bounded in magnitude by 10. In our nomen... |

2 |
Applied and Computational Complex Analysis: Special Functions
- Henrici
- 1988
(Show Context)
Citation Context ...ory of Laguerre asymptotics. 1.1 Research motives A primary research motive for providing effective asymptotics lies in a beautiful Laguerre series for the incomplete gamma function (see [1]), namely =-=[14]-=- Γ(a, z) = z a e −z 1 = ∞� n=0 (1 − a)n (n + 1)! 1 − a z + 1 1 + 2 − a z + 1 + · · · 1 L (−a) n (−z) L (−a) n+1 (−z), where (c)n := c(c + 1) · · · (c + n − 1) is the Pochhammer symbol. This series is ... |

2 |
Why steepest descents
- Olver
- 1970
(Show Context)
Citation Context ...nciple how to establish effective error bounds. The excellent treatment of effectiveness for Laplace’s method of steepest descent in [22] is a shining example. Even more illuminating is Olver’s paper =-=[21]-=-, which explains effective bounding and shows how unwieldy rigorous bounds can be. However, efficient algorithms for generating explicit effective big-O constants have only become practicable in recen... |

2 |
private communication
- Paris
- 2007
(Show Context)
Citation Context ...-arc series (64) errors for Bessel evaluation In(10); n = 4. The Hadamard series error behaves as K −n−1/2 while the exp-arc error behaves as K −n−1/2 2 −K (which is geometrical (linear) convergence) =-=[23]-=-. Though both series are absolutely convergent, here are some important differences between this Hadamard expansion and our exp-arc forms (64, 66). For example we have given our convergent sum only fo... |

2 |
On the use of Hadamard expansions in hyperasymptotic evaluation of Laplace-type integrals
- Paris
(Show Context)
Citation Context ...hole research area of convergent expansions related to classical, asymptotic ones has been pioneered in large part by R. Paris, whose works cover real and complex domains, saddle points, and the like =-=[24]-=- [25] [26] [27]. We should point out that Paris was able to develop within the last decade some similar, linearly convergent Bessel series by modifying the “tails” of Hadamard-class series. One open r... |

2 |
Assche, “Weighted zero distribution for polynomials orthogonal on an infinite interval
- VAN
- 1985
(Show Context)
Citation Context ...arcsin series development, as described in Section 5. There is an interesting anecdote that reveals the difficulty inherent in Laguerre asymptotics. Namely, W. Van Assche in an interesting 1985 paper =-=[34]-=- used the expansion (7) for work on zero-distributions, only to find by 2001 that the C1 term in that 1985 paper had been calculated incorrectly. The amended series is given in his correction note [35... |

2 |
Assche, Erratum to “Weighted zero distribution for polynomials orthogonal on an infinite interval
- van
- 2001
(Show Context)
Citation Context ...34] used the expansion (7) for work on zero-distributions, only to find by 2001 that the C1 term in that 1985 paper had been calculated incorrectly. The amended series is given in his correction note =-=[35]-=- as L (−a) n (−z) ∼ e−z/2 2 √ π e2√nz z1/4−a/2 � . 1 + n1/4+a/2 � 3 − 12a 2 + 24(1 − a)z + 4z 2 6 48 √ z � 1 √n + O � �� 1 , nsor in our own notation with m := n + 1, � � 3 − 12a2 − 24(1 + a)z + 4z2 ∼... |

2 | Computational methods in physics and engineering - Wong - 1997 |

2 |
Exactification of the method of steepest descents: The Bessel functions of large order and argument
- Paris
- 2004
(Show Context)
Citation Context ...f convergent expansions related to classical, asymptotic ones has been pioneered in large part by R. Paris, whose works cover real and complex domains, saddle points, and the like [29] [30] [31] [32] =-=[33]-=- [34]. We should point out that Paris was able to develop within the last decade some similar, linearly convergent Bessel series by modifying the “tails” of Hadamard-class series. Finally,we note that... |

2 |
Hadamard expansions for integrals with saddles coalescing with an endpoint
- Paris, Kaminski
- 2005
(Show Context)
Citation Context ...vergent expansions related to classical, asymptotic ones has been pioneered in large part by R. Paris, whose works cover real and complex domains, saddle points, and the like [29] [30] [31] [32] [33] =-=[34]-=-. We should point out that Paris was able to develop within the last decade some similar, linearly convergent Bessel series by modifying the “tails” of Hadamard-class series. Finally,we note that the ... |

2 |
private communication
- Temme
- 2006
(Show Context)
Citation Context ...n χ) , (50) k=0 8 Our oscillatory asymptotic (48) has been verified to several terms by N. Temme; also he verifies our claim that the classical Szegö coefficients do vanish for the parities indicated =-=[39]-=-. 24swith χ := z − πn/2 − π/4, bk := Bk(iz)e iπ/4 + Bk(−iz)e −iπ/4 , ick := Bk(iz)e iπ/4 − Bk(−iz)e −iπ/4 . Note that if z is real then each bk, ck is real, whence our series here has all real terms. ... |

2 | Linear difference equations with transition points
- Wang, Wong
- 2005
(Show Context)
Citation Context ...ions amongst the L (−a) n (−z)? One may ask the same question for the Laguerre differential equation as starting point. A promising research avenue for a discrete-iterative approach to asymptotics is =-=[45]-=-; see also [8] for the asymptotic analysis of certain complex continued fractions. As for differentialequation approaches, there is the classical work of Erdélyi and Olver, plus modern work on combina... |

1 |
Oval convergence regions for continued fractions K(an/1),” preprint
- Jacobsen, Thron
- 1986
(Show Context)
Citation Context ... accrue. Incidentally, we are aware that sub-exponential convergence results for the general incomplete-gamma-fraction might be attainable via the very complicated, seminal work of Jacobsen and Thron =-=[15]-=- on oval convergence regions, but at least our effective-Laguerre approach eventually yields the desired growth properties. We should state an important caveat at this juncture: We are not intending t... |

1 |
Inequalities for Laguerre functions
- Love
- 1997
(Show Context)
Citation Context ...ons. An important point here is that any asymptotic theory with n ≈ a would have to take into account these useful special cases. On the subject of rigorous bounds, there is the interesting treatment =-=[19]-=- in which L (µ) ν (x) is given upper (and lower!) bounds, for real x and ℜ(µ) > −1. One might call such results effective error bounds of zero-th order; in any case, they do not help the present treat... |

1 |
A new asymptotic form for the Laguerre polynomials
- Smith
- 1992
(Show Context)
Citation Context ... paper—involves asymptotic behavior when the subindex n is linked either to a or z, or both. For example, the paper [17] provides a large-n asymptotic expansion for L (a) n (nx). Another treatment is =-=[29]-=-, in which the author handles L (nx) n (ny). In [8] the authors analyze cases of L (an) n for which an >> n. One hopes that our present methods for effective bounding can be applied to such variants. ... |

1 |
Computing the incomplete gamma function to arbitrary precision
- Winitski
- 2003
(Show Context)
Citation Context ... applying the recurrence (10) in the next section to the generating function � n rntn . A more modern literature treatment that is again consistent with the heuristic (3)–(4), is given by Wipitski in =-=[37]-=-, where (8) is invoked to yield a contour integral for L (−a) n (−z). Then a stationaryphase approach yields precisely the correct asymptotic, at least for certain subregions of D. It should be noted ... |

1 |
Liner difference equations with transition points
- Wang, Wong
- 2004
(Show Context)
Citation Context ...t for the current authors. One may ask the same question for the Laguerre differential equation (12) as starting point. A promising research avenue for a discrete-iterative approach to asymptotics is =-=[38]-=-. • It would be useful to establish the very most efficient way to calculate Jn(z) with our converging series (66) and to know, for given arguments n, z how many terms of the exp-arc sum yield b good ... |

1 | The evaluation of Bessel functions via exp-arc integrals
- Borwein, Borwein, et al.
- 2007
(Show Context)
Citation Context ...s. Finally,we note that the problem of generalizing such unconditionally convergent series as our (50) to non-integer indices—and with comparison to the recent work of Paris—is analyzed in a new work =-=[3]-=-. 8 Open problems • How might one proceed with the exp-arc theory to obtain effective error bounds for oscillatory Laguerre modes, and-or oscillatory Bessel modes? We know that previous researchers ha... |

1 | Integer powers of Arcsin
- Borwein, Chamberland
- 2007
(Show Context)
Citation Context ...ials of the exp-arc method Now we investigate what we call exponential-arcsine (“exp-arc”) series. First, for any complex τ and x ∈ [−1, 1], one has a remarkable, absolutely convergent expansion (see =-=[5]-=-): e τ arcsin x = ∞� k=0 where the coefficients depend on the parity of the index: r2m+1(τ) := τ j=1 By differentiating with respect to x we obtain rk(τ) xk , (30) k! m� � 2 2 τ + (2j − 1) � m� � 2 2 ... |

1 |
to the Riemann–Siegel formula,” preprint
- Crandall, “Alternatives
- 2006
(Show Context)
Citation Context ...iemann himself [41, p. 22]. But when s has large imaginary height, one becomes interested in the incomplete gamma’s corresponding complex parameters (a, z) also of possibly large imaginary height [10]=-=[11]-=-. For example, in the art of prime-number counting, say for primes < 10 20 , one might need to know such as Γ � 3/4 + 10 10 i, z � , for various z also of large imaginary height, to good precision, in... |

1 |
A class of exactly soluble three-body problems
- Whitnell
- 1985
(Show Context)
Citation Context ...ations of Laguerre asymptotics. The Laguerre functions appear standardly in the quantum theory of the hydrogen atom [47, Ch. 4] and in certain exactly solvable three-body problems of chemical physics =-=[12]-=-[13]. The so-called WKB phase of a quantum eigenstate can, in such cases and for high quantum numbers, be calculated via Laguerre asymptotics. The Hermite polynomials Hn(x)—closely related to the Lagu... |

1 |
Exactly soluble two-electron atomic model
- Bettega
- 1984
(Show Context)
Citation Context ...ns of Laguerre asymptotics. The Laguerre functions appear standardly in the quantum theory of the hydrogen atom [47, Ch. 4] and in certain exactly solvable three-body problems of chemical physics [12]=-=[13]-=-. The so-called WKB phase of a quantum eigenstate can, in such cases and for high quantum numbers, be calculated via Laguerre asymptotics. The Hermite polynomials Hn(x)—closely related to the Laguerre... |

1 | A Treatise on Plane Trigonometry, London,Cambridge - Hobson - 1925 |