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Fo(p)1260 3731 y Fk(\000)20 b Fw(\()p Fq(p)1432 3694 y Fo(\027)1495 3731 y Fw(+)g Fq(p)1632 3694 y Fj(\000)p Fo(\027)1730 3731 y Fw(\))p Fq(\033)1817 3745 y Fo(p)1877 3731 y Fw(+)g(\(1)h(+)f Fq(x)2212 3694 y Ft(2)2251 3731 y Fq(=)p Fw(4\))2376 3658 y Fi(\001)2434 3731 y Fq(y)s Fw(\()p Fq(x)p Fw(\))26 b(=)f(0)p Fq(:)0 3917 y Fw(By)31 b(c)m(hanging)f Fq(x)g Fw(in)m(to)h(1)p Fq(=t)g Fw(and)e Fq(y)s Fw(\()p Fq(x)p Fw(\))i(in)m(to)g Fq(z)t Fw(\(1)p Fq(=t)p Fw(\),)h(the)e(equation)h(b)s(ecomes)1026 4024 y Fi(\022)1093 4052 y(\020)1148 4153 y Fw(1)20 b(+)1393 4091 y(1)p 1314 4132 203 4 v 1314 4215 a(4)p Fq(p)1405 4189 y Ft(4)1445 4215 y Fq(t)1478 4189 y Ft(2)1527 4052 y Fi(\021)1581 4153 y Fq(\033)1636 4115 y Ft(2)1633 4175 y Fo(p)1696 4153 y Fk(\000)g Fw(\()p Fq(p)1868 4115 y Fo(\027)1932 4153 y Fw(+)f Fq(p)2068 4115 y Fj(\000)p Fo(\027)2166 4153 y Fw(\))p Fq(\033)2253 4167 y Fo(p)2313 4153 y Fw(+)h(1)2449 4024 y Fi(\023)2532 4153 y Fq(z)t Fw(\()p Fq(t)p Fw(\))26 b(=)f(0)p Fq(:)0 4389 y Fw(The)i(exp)s(onen)m (tial)g(part)g(of)h(the)f(b)s(eha)m(viour)g(\(see)h(Eq.)g(\(1\)\))h(is) d(sough)m(t)i(in)f(terms)g(of)h Fk(E)2937 4403 y Fo(\013)2986 4389 y Fw(\()p Fq(t)p Fw(\))e(=)f(1)p Fq(=\022)3344 4403 y Fo(p)3384 4389 y Fw(\()p Fk(\000)p Fq(\013t)p Fw(\),)k(whic)m(h)0 4497 y(is)g(cancelled)h(b)m(y)h Fq(\033)659 4511 y Fo(p)718 4497 y Fw(+)20 b Fq(\013t)p Fw(.)41 b(The)30 b(c)m(hange)i(of)e(unkno)m (wn)f(function)g Fq(z)t Fw(\()p Fq(t)p Fw(\))d(=)f Fk(E)2621 4511 y Fo(\013)2671 4497 y Fw(\()p Fq(t)p Fw(\))p Fq(f)2819 4511 y Fo(\013)2868 4497 y Fw(\()p Fq(t)p Fw(\))31 b(leads)f(to)849 4608 y Fi(\022)916 4636 y(\020)970 4736 y Fw(1)21 b(+)1215 4675 y(1)p 1137 4716 V 1137 4799 a(4)p Fq(p)1228 4773 y Ft(4)1267 4799 y Fq(t)1300 4773 y Ft(2)1350 4636 y Fi(\021)1404 4736 y Fq(\013)1462 4699 y Ft(2)1502 4736 y Fq(pt)1581 4699 y Ft(2)1620 4736 y Fq(\033)1675 4699 y Ft(2)1672 4759 y Fo(p)1735 4736 y Fk(\000)f Fq(\013t)p Fw(\()p Fq(p)1998 4699 y Fo(\027)2061 4736 y Fw(+)g Fq(p)2198 4699 y Fj(\000)p Fo(\027)2296 4736 y Fw(\))p Fq(\033)2383 4750 y Fo(p)2443 4736 y Fk(\000)g Fw(1)2579 4608 y Fi(\023)2662 4736 y Fq(f)2707 4750 y Fo(\013)2756 4736 y Fw(\()p Fq(t)p Fw(\))25 b(=)g(0)p Fq(:)0 4980 y Fw(Th)m(us,)31 b(b)m(y)g(c)m(ho)s (osing)g Fq(\013)h Fw(suc)m(h)f(that)h Fq(\013)1306 4947 y Ft(2)1372 4980 y Fw(=)27 b(4)p Fq(p)1561 4947 y Ft(3)1600 4980 y Fw(,)32 b(one)g(gets)g(an)f(equation)g(for)g Fq(f)2697 4994 y Fo(\013)2778 4980 y Fw(whic)m(h)f(has)h(p)s(o)m(w)m(er)g(series) g(solu-)0 5088 y(tions.)40 b(A)31 b(further)e(simpli\014cation)e(is)i (ac)m(hiev)m(ed)i(b)m(y)g(considering)d(the)j(\\)p Fq(p)p Fw(-Borel)g(transform")f(of)g(the)h(series)e Fq(f)3814 5102 y Fo(\013)3863 5088 y Fw(:)0 5283 y(\(5\))1052 b Fq(g)1210 5297 y Fo(\013)1259 5283 y Fw(\()p Fq(\034)10 b Fw(\))26 b(:=)g Fk(B)1587 5297 y Fo(p)1626 5283 y Fq(f)1671 5297 y Fo(\013)1720 5283 y Fw(\()p Fq(\034)10 b Fw(\))26 b(=)1963 5196 y Fi(X)1962 5392 y Fo(n)p Fj(\025)p Ft(0)2110 5283 y Fq(a)2158 5297 y Fo(n)2205 5283 y Fq(p)2251 5245 y Fj(\000)p Fo(n)p Ft(\()p Fo(n)p Fj(\000)p Ft(1\))p Fo(=)p Ft(2)2611 5283 y Fq(\034)2661 5245 y Fo(n)2708 5283 y Fq(;)p eop %%Page: 144 4 144 3 bop 0 66 a Ft(144)107 b Fg(On)23 b(Jac)n(kson's)i Fo(q)r Fg(-Bessel)e(F)-6 b(unctions)0 266 y Fw(where)34 b Fq(a)315 280 y Fo(n)397 266 y Fw(are)h(the)g(co)s(e\016cien)m(ts)h (of)e Fq(f)1331 280 y Fo(\013)1380 266 y Fw(.)54 b(By)35 b(the)g(comm)m(utation)h(rule)d Fk(B)2571 280 y Fo(p)2610 266 y Fw(\()p Fq(t)2678 233 y Fo(m)2745 266 y Fq(\033)2800 233 y Fo(`)2797 288 y(p)2837 266 y Fw(\))g(=)f Fq(p)3054 233 y Fj(\000)p Fo(m)p Ft(\()p Fo(m)p Fj(\000)p Ft(1\))p Fo(=)p Ft(2)3453 266 y Fq(\034)3503 233 y Fo(m)3570 266 y Fq(\033)3625 233 y Fo(`)p Fj(\000)p Fo(m)3622 288 y(p)3775 266 y Fk(B)3835 280 y Fo(p)3875 266 y Fw(,)0 374 y Fq(g)43 388 y Fo(\013)123 374 y Fw(is)e(solution)e(of)j(a)g(t)m(w)m(o-term)h Fq(q)s Fw(-di\013erence)e(equation.)40 b(This)29 b(is)g(easily)h(solv)m (ed)g(to)h(\014nd)1235 557 y Fq(g)1278 571 y Fo(\013)1328 557 y Fw(\()p Fq(\034)10 b Fw(\))26 b(=)2082 496 y(1)p 1579 537 1051 4 v 1579 620 a(\()p Fk(\000)p Fq(\013p)1789 594 y Fo(\027)1833 620 y Fq(\034)10 b Fw(;)15 b Fq(q)s Fw(\))2002 634 y Fj(1)2077 620 y Fw(\()p Fk(\000)p Fq(\013p)2287 594 y Fj(\000)p Fo(\027)2386 620 y Fq(\034)10 b Fw(;)15 b Fq(q)s Fw(\))2555 634 y Fj(1)2640 557 y Fq(:)0 764 y Fw(It)32 b(follo)m(ws)f(that)i Fq(g)642 778 y Fo(\013)724 764 y Fw(is)e(meromorphic)g(in)g Fe(C)56 b Fw(with)30 b(\(simple\))h(p)s(oles)g(at)i Fk(f\000)p Fq(p)2647 731 y Fo(\027)t Fj(\000)p Ft(2)p Fo(n)2823 764 y Fq(=\013;)15 b Fk(\000)p Fq(p)3083 731 y Fj(\000)p Fo(\027)t Fj(\000)p Ft(2)p Fo(n)3315 764 y Fq(=\013)p Fk(g)34 b Fw(for)e Fq(n)27 b Fk(2)h Fe(N)7 b Fw(,)0 872 y(whic)m(h)29 b(implies)f(that)j Fq(f)811 886 y Fo(\013)890 872 y Fw(is)e(an)i(en)m(tire)f(function.)100 980 y(In)d(order)g(to)h(reco)m(v)m(er)h Fq(f)907 994 y Fo(\013)984 980 y Fw(from)e Fq(g)1239 994 y Fo(\013)1289 980 y Fw(,)h(the)g Fq(p)p Fw(-Borel)g(transform)f(of)h(\(5\))g(is)f (rev)m(erted)h(b)m(y)g(means)f(of)h(a)g(Hadamard)0 1088 y(pro)s(duct)k(of)h Fq(g)493 1102 y Fo(\013)577 1088 y Fw(with)e Fq(\022)829 1102 y Fo(p)869 1088 y Fw(.)49 b(This)32 b(leads)g(to)i(a)g(Cauc)m(h)m(y)g(in)m(tegral)f(represen)m (tation)g(from)g(whic)m(h)f(a)h(residue)f(com-)0 1196 y(putation)e(yields)e(the)j(connection)f(form)m(ula.)40 b(The)30 b(Cauc)m(h)m(y)h(in)m(tegral)f(is)1270 1384 y Fq(f)1315 1398 y Fo(\013)1364 1384 y Fw(\()p Fq(t)p Fw(\))c(=)1642 1323 y(1)p 1599 1364 132 4 v 1599 1447 a(2)p Fq(\031)s(i)1756 1261 y Fi(Z)1806 1467 y Fj(j)p Fo(\034)8 b Fj(j)p Ft(=)p Fo(r)1992 1384 y Fq(g)2035 1398 y Fo(\013)2085 1384 y Fw(\()p Fq(\034)i Fw(\))p Fq(\022)2248 1398 y Fo(p)2288 1384 y Fw(\()p Fq(t=\034)g Fw(\))2496 1323 y Fq(d\034)p 2496 1364 98 4 v 2521 1447 a(\034)2605 1384 y(;)0 1607 y Fw(where)29 b Fq(r)e(<)e Fw(min)578 1533 y Fi(\000)619 1607 y Fk(j)p Fq(p)690 1574 y Fo(\027)734 1607 y Fq(=\013)p Fk(j)p Fq(;)15 b Fk(j)p Fq(p)973 1574 y Fj(\000)p Fo(\027)1072 1607 y Fq(=\013)p Fk(j)1200 1533 y Fi(\001)1243 1607 y Fw(.)40 b(The)29 b(only)f(residues)f(come)j(from)f(the)g(p)s(oles)f(of)h Fq(g)3009 1621 y Fo(\013)3059 1607 y Fw(.)40 b(The)29 b(asymptotic)g(b)s(e-)0 1715 y(ha)m(viour)22 b(of)g Fq(g)459 1729 y Fo(\013)532 1715 y Fw(implies)d(that)24 b(this)d(in)m(tegral)h (is)g(equal)g(to)h(the)g(sum)e(of)i(the)g(residues)e(and)h(an)g(actual) h(computation)0 1823 y(of)31 b(these)f(residues)f(leads)h(to)127 2014 y Fq(f)172 2028 y Fo(\013)221 2014 y Fw(\()p Fq(t)p Fw(\))c(=)542 1952 y Fq(\022)585 1966 y Fo(p)624 1952 y Fw(\()p Fk(\000)p Fq(\013q)832 1919 y Fo(\027)t(=)p Ft(2)946 1952 y Fq(t)p Fw(\))p 456 1993 646 4 v 456 2076 a(\()p Fq(q)s Fw(;)15 b Fq(q)s Fw(\))654 2090 y Fj(1)729 2076 y Fw(\()p Fq(q)808 2050 y Fj(\000)p Fo(\027)907 2076 y Fw(;)g Fq(q)s Fw(\))1026 2090 y Fj(1)1111 2028 y Ft(2)1150 2014 y Fq(\036)1204 2028 y Ft(1)1244 2014 y Fw(\(0)p Fq(;)g Fw(0;)g Fq(q)1493 1976 y Fo(\027)t Ft(+1)1628 2014 y Fw(;)g Fq(q)s(;)g Fk(\000)p Fq(x)1875 1976 y Ft(2)1915 2014 y Fq(=)p Fw(4\))21 b(+)2194 1952 y Fq(\022)2237 1966 y Fo(p)2276 1952 y Fw(\()p Fk(\000)p Fq(\013q)2484 1919 y Fj(\000)p Fo(\027)t(=)p Ft(2)2653 1952 y Fq(t)p Fw(\))p 2162 1993 591 4 v 2162 2076 a(\()p Fq(q)s Fw(;)15 b Fq(q)s Fw(\))2360 2090 y Fj(1)2436 2076 y Fw(\()p Fq(q)2515 2050 y Fo(\027)2558 2076 y Fw(;)g Fq(q)s Fw(\))2677 2090 y Fj(1)2762 2028 y Ft(2)2802 2014 y Fq(\036)2856 2028 y Ft(1)2895 2014 y Fw(\(0)p Fq(;)g Fw(0;)g Fq(q)3144 1976 y Fj(\000)p Fo(\027)t Ft(+1)3335 2014 y Fw(;)g Fq(q)s(;)g Fk(\000)p Fq(x)3582 1976 y Ft(2)3621 2014 y Fq(=)p Fw(4\))p Fq(;)0 2213 y Fw(where)32 b Fq(xt)e Fw(=)f(1)k(and)f Fk(j)p Fq(x)p Fk(j)e Fq(<)g Fw(2.)48 b(With)33 b(v)m(ery)g(little)f(rewriting,)g(this)f(is)h(the)h(desired)f (connection)h(form)m(ula.)47 b(The)0 2321 y(limiting)27 b(b)s(eha)m(viour)i(of)i(this)e(form)m(ula)h(when)f Fq(q)f Fk(!)d Fw(1)31 b(is)f(studied)e(in)h([5)q(].)100 2429 y(The)23 b(second)h(family)e(of)i Fq(q)s 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Fw(in)e(the)i(de\014nition)d(of)j Fq(J)3283 2853 y Ft(\(1\))3274 2912 y Fo(\027)3409 2901 y Fw(is)f(the)3660 2915 y Ft(0)3699 2901 y Fq(\036)3753 2915 y Ft(1)3824 2901 y Fw(in)0 3024 y(that)c(of)h Fq(J)353 2976 y Ft(\(2\))344 3036 y Fo(\027)447 3024 y Fw(.)40 b(F)-8 b(rom)27 b(there,)h(a)f(Cauc)m (h)m(y)h(in)m(tegral)e(represen)m(tation)h(follo)m(ws)f(and)g(again)h (a)h(residue)d(computation)0 3132 y(giv)m(es)h(the)f(connection)h(form) m(ula,)g(thanks)f(to)h(extra)g(considerations)f(ab)s(out)g(the)g (asymptotic)h(b)s(eha)m(viour)e(of)i(the)0 3240 y(in)m(tegrand.)1648 3424 y(5.)47 b Fv(Commen)m(ts)100 3586 y Fw(It)37 b(has)g(b)s(een)g (observ)m(ed)g(that)h(the)g(connection)g(\\constan)m(ts")h(p)s(ossess)e (the)g(nice)g(prop)s(ert)m(y)g(that)h(they)g(are)0 3693 y(elliptic)e(in)h(the)i(case)h(of)e(Heine's)h(function.)63 b(This)37 b(is)h(a)g(general)h(phenomenon)e([3)q(].)65 b(The)38 b(form)m(ulae)h(in)e(the)0 3801 y Fq(q)s Fw(-w)m(orld)d(imply) f(imp)s(ortan)m(t)h(iden)m(tities)g(\(after)i(all,)g(Jacobi's)f(triple) e(pro)s(duct)h(can)i(b)s(e)e(seen)i(as)f(a)h(connection)0 3909 y(form)m(ula\).)j(Recen)m(t)28 b(w)m(ork)e(b)m(y)g(Changgui)f (Zhang)g(sho)m(ws)h(that)h(the)g(limiting)c(b)s(eha)m(viour)i(of)h (these)h Fq(q)s Fw(-connection)0 4017 y(form)m(ulae)j(when)f Fq(q)f Fk(!)e Fw(1)k(yields)f(the)i(Stok)m(es)g(phenomenon)e(of)i(the)f (di\013eren)m(tial)f(w)m(orld.)1682 4192 y Fs(Bibliograph)m(y)0 4338 y Fy([1])39 b(Andrews)31 b(\(George)h(E.\),)h(Ask)n(ey)c(\(Ric)n (hard\),)j(and)e(Ro)n(y)g(\(Ranjan\).)h({)g Fu(Sp)l(e)l(cial)h (functions)p Fy(.)g({)f(Cam)n(bridge)g(Univ)n(ersit)n(y)f(Press,)119 4429 y(Cam)n(bridge,)c Fd(1999)r Fy(,)g(xvi+664p.)0 4520 y([2])39 b(Gasp)r(er)24 b(\(George\))f(and)f(Rahman)e(\(Mizan\).)k({)e Fu(Basic)j(hyp)l(er)l(ge)l(ometric)i(series)p Fy(.)d({)e(Cam)n(bridge)g (Univ)n(ersit)n(y)f(Press,)k(Cam)n(bridge,)119 4612 y Fd(1990)r Fy(,)h(xx+287p.)g(With)f(a)h(forew)n(ord)h(b)n(y)e(Ric)n (hard)g(Ask)n(ey)-6 b(.)0 4703 y([3])39 b(Saulo)n(y)27 b(\(Jacques\).)g({)g(Syst)n(\022)-36 b(emes)26 b(aux)g Fd(q)s Fy(-di\013)n(\023)-36 b(erences)26 b(singuliers)j(r)n(\023)-36 b(eguliers)28 b(:)36 b(classi\014cation,)29 b(matrice)e(de)f(connexion) h(et)f(mon-)119 4794 y(o)r(dromie.)h Fu(A)n(nnales)h(de)g(l'Institut)f (F)-6 b(ourier)p Fy(,)27 b(v)n(ol.)f(50,)h(n)1737 4800 y(\027)1794 4794 y(4,)g Fd(2000)q Fy(,)f(pp.)f(1021{1071.)0 4886 y([4])39 b(Slater)d(\(Lucy)e(Joan\).)i({)f Fu(Gener)l(alize)l(d)i (hyp)l(er)l(ge)l(ometric)i(functions)p Fy(.)d({)g(Cam)n(bridge)e(Univ)n (ersit)n(y)g(Press,)39 b(Cam)n(bridge,)e Fd(1966)r Fy(,)119 4977 y(xiii+273p.)0 5068 y([5])i(Zhang)32 b(\(Changgui\).)h({)f Fu(Sur)i(la)f(sommabilit)n(\023)-37 b(e)33 b(des)h(s)n(\023)-37 b(eries)34 b(enti)n(\022)-37 b(er)l(es)35 b(solutions)f(formel)t(les)f (des)f(\023)-37 b(equations)35 b(aux)f Fd(q)s Fu(-di\013)n(\023)-37 b(er)l(enc)l(es)119 5160 y(lin)n(\023)g(eair)l(es)28 b(et)g(\022)-39 b(a)28 b(c)l(o)l(e\016cients)h(analytiques)p Fy(.)f({)e(M)n(\023)-36 b(emoire)26 b(d'habilitation,)h(Univ)n(ersit)n (\023)-36 b(e)25 b(de)h(La)g(Ro)r(c)n(helle,)g Fd(2000)r Fy(.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF/tvz@princeton.edu