We consider the hypergeometric equation (1 − t)∂t∂t∂g + x3g = 0, whose unique analytic solution φ1(t; x) = 1 + O(t) near t = 0 for t = 1 becomes a generating function for multiple zeta values φ1(1; x) = f3(x) = 1 − ζ(3)x3 + ζ(3, 3)x6 − …. We apply the so-called WKB method to study solutions of the hypergeometric equation for large x and we calculate corresponding Stokes matrices. We prove that the function f3(x) near x = ∞ is also expressed via WKB type functions which subject to some Stokes phenomenon. This implies that f3(x) satisfies a sixth order linear differential equation with irregular singularity at infinity.
REFERENCES
1.
Aoki
, T.
, Ohno
, Y.
, and Wakabayashi
, N.
, “On generating functions of multiple zeta values and generalized hypergeometric functions
,” Manuscripta Math.
134
, 139
–155
(2011
).2.
3.
Balser
, W.
, in From Divergent Power Series to Analytic Functions
, Lecture Notes in Mathematics
, Vol. 1582
(Springer–Verlag
, Berlin
, 1994
).4.
Bateman
, G.
and Erdelyi
, A.
, Higher Transcendental Functions
(McGraw Hill
, New York
, 1953
), Vol. 1.5.
Berk
, H. L.
, Nevins
, W. M.
, and Robert
, K. V.
, “New Stokers' line in WKB theory
,” J. Math. Phys.
23
, 988
–1002
(1982
).6.
Beukers
, F.
, Bornawell
, D.
, and Heckman
, G.
, “Siegel normality
,” Ann. Math.
127
, 279
–308
(1988
).7.
Beukers
, F.
and Heckman
, G.
, “Monodromy for the hypergeometric function nFn − 1
,” Invent. Math.
95
, 325
–354
(1989
).8.
Beukers
, F.
and Peters
, C. A. M.
, “A family of K3 surfaces and ζ(3)
,” J. Reine Angew. Math.
351
, 42
–54
(1984
).9.
Birkhoff
, G. D.
, “On the asymptotic character of the solutions of certain differential equations containing a parameter
,” Trans. Am. Math. Soc.
9
, 219
–230
(1908
).10.
Borwein
, J. M.
, Bradley
, D. M.
, and Broadhurst
, D. J.
, “Evaluations of |$\mathit {k-}$|fold Euler/Zagier sums: A compendium of results for arbitrary 𝓀
,” Electron. J. Comb.
4
(2
), Res. Paper 5, 21pp
(1997
).11.
Borwein
, J. M.
, Bradley
, D. M.
, Broadhurst
, D. J.
, and Lisoněk
, P.
, “Combinatorial aspects of multiple zeta values
,” Electron. J. Comb.
5
, Res. Paper 38, 12pp
(1998
).12.
Cresson
, J.
, Fischler
, S.
, and Rivoal
, T.
, “Séries hypergéométriques multiples et polyzêtas
,” Bull. Soc. Math. France
136
, 97
–145
(2008
).13.
Delabaere
, E.
and Pham
, F.
, “Resurgent methods in semi-classical asymptotics
,” Ann. Inst. Henri Poincaré. Phys. Théoret.
71
, 1
–94
(1999
).14.
Duval
, A.
and Mitschi
, C.
, “Matrices de Stokes et groupe de Galois des équations hypergéométriques confluentes generalizes
,” Pacific J. Math.
138
, 25
–56
(1989
).15.
Fedoryuk
, M. B.
, Asymptotic Analysis. Linear Ordinary Differential Equations
(Springer–Verlag
, Berlin
, 1993
);Russian:
Asymptotic Methods for Linear Ordinary Differential Equations
(Nauka
, Moscow
, 1983
).16.
Glyutsuk
, A. A.
, “Stokes operators via limit monodromy of generic perturbation
,” J. Dynam. Control Syst.
5
, 101
–135
(1999
).17.
18.
Katz
, N. M.
, “On the calculation of some differential Galois groups
,” Invent. Math.
87
, 13
–61
(1987
).19.
Katz
, N. M.
, in “Exponential sums and differential equations
,” Annals of Mathematical Studies
, Vol. 124 (Princeton University Press
, Princeton
, 1990
).20.
Kohno
, M.
, in “Global analysis in linear differential equations
,” Mathematics in Applications
(Kluwer Academic
, Dordrecht
, 1999
).21.
Kontsevich
, M.
and Zagier
, D.
, in “Periods
,” Mathematics Unlimited – 2001 and Beyond
(Springer–Verlag
, Berlin
, 2001
), pp. 771
–808
.22.
Li
, Z. -H.
, “Sum of multiple zeta values of fixed weight, depth and i–height
,” Math. Zeitschrift
258
, 133
–142
(2008
).23.
Lian
, B. H.
and Yau
, S. -T.
, “Arithmetic properties of mirror map and quantum coupling
,” Commun. Math. Phys.
176
, 163
–291
(1996
).24.
25.
Minh
, H. N.
, Petitot
, M.
, and van der Hoeven
, J.
, “Formal Series and Algebraic Combinatorics, Toronto 98
,” Discrete Math.
225
, 217
–230
(2000
).26.
Mitschi
, C.
, “Differential Galois groups of confluent generalized hypergeometric equations: an approach using Stokes multipliers
,” Pacific J. Math.
176
, 365
–405
(1996
).27.
Ohno
, Y.
and Zagier
, D.
, “Multiple zeta values of fixed weight, depth and weight
,” Indag. Math.
12
, 483
–487
(2001
).28.
van der Poorten
, A.
, “A proof that Euler missed... Apéry's proof of the irrationality of ζ(3)
,” Math. Intell.
1
, No 4, 195
–2003
(1978/79
).29.
Ramis
, J. -P.
, “Séries divergentes et théories asymptotiques
,” Panoramas et Synthesèses, Soc. Math. de France, 1993
.30.
31.
Wasow
, W.
, Asymptotic Expansions for Ordinary Differential Equations
, (Wiley
, New York
, 1965
).32.
Zagier
, D.
, “Values of zeta function and their applications
,” in First European Congress of Mathematicians
, Progress in Mathematics
Vol. 120
(Birkhaüser
, Basel
, 1994
), Vol. 2, pp. 497
–512
.33.
Zakrzewski
, M.
and Żołądek
, H.
, “Linear differential equations and multiple zeta values. I. Zeta(2)
,” Fundam. Math.
210
, 207
–242
(2010
).34.
Zakrzewski
, M.
and Żołądek
, H.
, “Linear differential equations and multiple zeta values. II. A generalization of the WKB method
,” J. Math. Anal. Appl.
383
, 55
–70
(2011
).35.
Zhao
, J.
, “On a conjecture of Borwein, Bradley and Broadhurst
,” J. Reine Angew. Math.
639
, 223
–233
(2010
).36.
Żołądek
, H.
, “The extended monodromy group and Liouvillian first integrals
,” J. Dynam. Control Syst.
4
, 1
–28
(1998
).37.
Żołądek
, H.
, “Note on multiple zeta–values
,” Bull. Acad. Ştiinţe Rep. Mold. Matem.
41
, 78
–82
(2003
).38.
39.
Zudilin
, V. V.
, “Arithmetic hypergeometric series
,” Russian Math. Surveys
66
, 369
–420
(2011
).40.
The deformation |$\left\lbrace \mathbb {T}_{\kappa }\right\rbrace _{0\le \kappa \le 1}$| can be given explicitly as follows: 0 ⩽ α, β ⩽ 2π. Here the modulus A(α) is a smooth function such that: A(α) = |Pσ| in δ–neighborhood of |$\alpha _{\sigma }=\arg P_{\sigma },$| A(α) = 1 outside 2δ–neighborhood of the critical set {ασ} and A(α) varies between 1 and |Pσ|; in the same way, the modulus B(β) is defined. It is easy that Pκ(α), Q κ(α, β) = [B2/A]κ/2ei(β − α/2) and Rκ(α, β) = [B2A]−κ/2e−i(β + α/2) avoid the singular value t−1/3, provided the constant δ is sufficiently small.
\begin{equation*}P_{\kappa }(\alpha )=A(\alpha )^{\kappa }e^{i\alpha },\text{ }S_{\kappa }(\beta )=B(\beta )^{\kappa }e^{i\beta },\end{equation*}
© 2012 American Institute of Physics.
2012
American Institute of Physics
You do not currently have access to this content.
