View source for G-function

The concept of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110010/g1100102.png" />-functions, not to be confused with the [[Meijer-G-functions|Meijer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110010/g1100103.png" />-functions]], was introduced by C.L. Siegel around 1929 [[#References|[a1]]], p. 239, in connection with transcendence questions. Consider a [[Taylor series|Taylor series]] of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110010/g1100104.png" /></td> </tr></table>

where the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110010/g1100105.png" /> all belong to a fixed algebraic number field (cf. also [[Algebraic number|Algebraic number]]; [[Field|Field]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110010/g1100106.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110010/g1100107.png" />). Suppose it satisfies the following conditions:

i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110010/g1100108.png" /> satisfies a linear differential equation (cf. also [[Linear differential operator|Linear differential operator]]) with polynomial coefficients;

ii) there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110010/g1100109.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110010/g11001010.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110010/g11001011.png" /> is called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110010/g11001012.png" />-function. Here, the notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110010/g11001013.png" /> stands for the so-called projective height, given by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110010/g11001014.png" /></td> </tr></table>

for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110010/g11001015.png" />-tuple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110010/g11001016.png" />. The product is taken over all valuations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110010/g11001017.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110010/g11001018.png" /> (cf. also [[Norm on a field|Norm on a field]]). When the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110010/g11001019.png" /> are rational numbers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110010/g11001020.png" /> is simply the maximum of the absolute values of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110010/g11001021.png" /> times their common denominator. Roughly speaking, Siegel's <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110010/g11001022.png" />-functions can be considered as (very interesting) variations on the geometric series, hence the name.

The most common examples are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110010/g11001023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110010/g11001024.png" />, algebraic functions, and the ordinary hypergeometric functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110010/g11001025.png" /> with rational parameters (cf. also [[Hypergeometric function|Hypergeometric function]]). Siegel introduced them along with their relatives, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110010/g11001026.png" />-functions (cf. [[E-function|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110010/g11001027.png" />-function]]), which can be considered as variations on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110010/g11001028.png" />.

Although Siegel states some irrationality results for values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110010/g11001029.png" />-functions at algebraic points, he never published the details of his computations. Subsequent work of A.I. Galoschkin and others showed that there are many more obstacles to get arithmetic results for values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110010/g11001030.png" />-functions than for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110010/g11001031.png" />-functions. Significant progress was achieved in the 1980{}s, notably by E. Bombieri [[#References|[a2]]] and G.V. Chudnovsky [[#References|[a3]]]. Much of this progress was related to the properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110010/g11001032.png" />-functions themselves and to the question of what <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110010/g11001033.png" />-functions really are. If one takes any linear differential equation with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110010/g11001034.png" />, its [[Power series|power series]] solutions will usually not be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110010/g11001035.png" />-functions. An introduction to these arithmetical problems is given in [[#References|[a4]]]. Having a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110010/g11001036.png" />-function solution poses considerable constraints on the arithmetic of a linear differential equation. There exist several conjectures in this direction, the most important being the Bombieri–Dwork conjecture that the differential equation should come from [[Algebraic geometry|algebraic geometry]] in a suitable sense. All known <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110010/g11001037.png" />-functions actually arise in this way. The converse statement is known to be true, see [[#References|[a5]]]. So, statements on the arithmetic nature of values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110010/g11001038.png" />-functions can also have consequences for problems in algebraic geometry. This point of view was notably adopted by Y. André [[#References|[a6]]].

Presently (1996), one can say that Siegel's method and its improvements have not yet given a transcendence proof for a single value of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110010/g11001039.png" />-function. The fact that such transcendence proofs are a subtle matter was once more confirmed by J. Wolfart's discovery of transcendental <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110010/g11001040.png" />-functions having algebraic values at algebraic points [[#References|[a7]]]. See also [[#References|[a8]]], [[#References|[a9]]] for actual computations both in the Archimedean and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110010/g11001041.png" />-adic domain.

See also [[E-function|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110010/g11001042.png" />-function]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> C.L. Siegel, "Über einige Anwendungen diophantischer Approximationen" , ''Ges. Abhandlungen'' , '''I''' , Springer (1966) {{MR|}} {{ZBL|56.0180.05}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> E. Bombieri, "On <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110010/g11001043.png" />-functions" , ''Recent Progress in Analytic Number Theory'' , '''2''' , Acad. Press (1981) pp. 1–67 {{MR|0185108}} {{ZBL|0136.06503}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> D. Chudnovski, G. Chudnovski, "Applications of Padé-approximations to diophantine inequalities in values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110010/g11001044.png" />-functions" , ''Lecture Notes in Mathematics'' , '''1052''' , Springer (1982) pp. 1–51</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> B. Dwork, G. Gerotto, F.J. Sullivan, "An introduction to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110010/g11001045.png" />-functions" , ''Ann. Math. Studies'' , '''133''' , Princeton Univ. Press (1994) {{MR|1274045}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> N.M. Katz, "On differential equations satisfied by period matrices" ''IHES Publ. Math.'' , '''35''' (1968) {{MR|}} {{ZBL|0159.22502}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> Y. André, "G-functions and geometry" , ''Aspects of Mathematics'' , Vieweg (1989) {{MR|0990016}} {{ZBL|0688.10032}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> J. Wolfart, "Werte hypergeometrischer Funktionen" ''Invent. Math.'' , '''92''' (1988) pp. 187–216 {{MR|0931211}} {{ZBL|0649.10022}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> F. Beukers, J. Wolfart, "Algebraic values of hypergeometric functions" , ''New Advances in Transcendence Theory'' , Cambridge Univ. Press (1988) {{MR|0971994}} {{ZBL|0656.10030}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> F. Beukers, "Algebraic values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110010/g11001046.png" />-functions" ''J. Reine Angew. Math.'' , '''434''' (1993) pp. 45–65 {{MR|1195690}} {{ZBL|}} </TD></TR></table>