2 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software
3 %%Title: tchebychev.dvi
6 %%BoundingBox: 0 0 596 842
7 %%DocumentPaperSizes: a4
9 %DVIPSCommandLine: dvips -Pljfour -o tchebychev.ps tchebychev.dvi
10 %DVIPSParameters: dpi=600, compressed, comments removed
11 %DVIPSSource: TeX output 1997.02.16:1406
12 %%BeginProcSet: texc.pro
13 /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N
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15 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1}
16 ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale
17 isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div
18 hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul
19 TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if}
20 forall round exch round exch]setmatrix}N /@landscape{/isls true N}B
21 /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B
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24 string /base X array /BitMaps X /BuildChar{CharBuilder}N /Encoding IE N
25 end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /df{
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27 N df-tail}B /E{pop nn dup definefont setfont}B /ch-width{ch-data dup
28 length 5 sub get}B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{
29 128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub
30 get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data
31 dup type /stringtype ne{ctr get /ctr ctr 1 add N}if}B /id 0 N /rw 0 N
32 /rc 0 N /gp 0 N /cp 0 N /G 0 N /sf 0 N /CharBuilder{save 3 1 roll S dup
33 /base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx
34 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff
35 setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff
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37 /cp 0 N{rc 0 ne{rc 1 sub /rc X rw}{G}ifelse}imagemask restore}B /G{{id
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40 /gp X adv}B /nd{/cp 0 N rw exit}B /lsh{rw cp 2 copy get dup 0 eq{pop 1}{
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42 adv}B /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2
43 idiv S 128 and or}ifelse}ifelse put 1 adv}B /clr{rw cp 2 index string
44 putinterval adv}B /set{rw cp fillstr 0 4 index getinterval putinterval
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46 {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{
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48 chg nd}{pop nd}]dup{bind pop}forall N /D{/cc X dup type /stringtype ne{]
49 }if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup
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51 cc 1 add D}B /bop{userdict /bop-hook known{bop-hook}if /SI save N @rigin
52 0 0 moveto /V matrix currentmatrix dup 1 get dup mul exch 0 get dup mul
53 add .99 lt{/QV}{/RV}ifelse load def pop pop}N /eop{SI restore userdict
54 /eop-hook known{eop-hook}if showpage}N /@start{userdict /start-hook
55 known{start-hook}if pop /VResolution X /Resolution X 1000 div /DVImag X
56 /IE 256 array N 0 1 255{IE S 1 string dup 0 3 index put cvn put}for
57 65781.76 div /vsize X 65781.76 div /hsize X}N /p{show}N /RMat[1 0 0 -1 0
58 0]N /BDot 260 string N /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V
59 {}B /RV statusdict begin /product where{pop product dup length 7 ge{0 7
60 getinterval dup(Display)eq exch 0 4 getinterval(NeXT)eq or}{pop false}
61 ifelse}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale rulex ruley false
62 RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR rulex ruley scale 1 1
63 false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave newpath transform
64 round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg
65 rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail
66 {dup /delta X 0 rmoveto}B /M{S p delta add tail}B /b{S p tail}B /c{-4 M}
67 B /d{-3 M}B /e{-2 M}B /f{-1 M}B /g{0 M}B /h{1 M}B /i{2 M}B /j{3 M}B /k{
68 4 M}B /w{0 rmoveto}B /l{p -4 w}B /m{p -3 w}B /n{p -2 w}B /o{p -1 w}B /q{
69 p 1 w}B /r{p 2 w}B /s{p 3 w}B /t{p 4 w}B /x{0 S rmoveto}B /y{3 2 roll p
70 a}B /bos{/SS save N}B /eos{SS restore}B end
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542 07804848131F00FF14FF90B6FCA2232B7DAA2B>I E end
545 %%Feature: *Resolution 600dpi
554 1 0 bop 404 573 a Fl(The)27 b(Tsc)m(heb)m(yc)m(hev)32
555 b(p)s(olynomials)23 b(\(of)k(the)g(1st)g(kind\))f Fk(T)2467
556 588 y Fj(n)2514 573 y Fl(\()p Fk(x)p Fl(\))h(are)g(de\014ned)h(through)
557 257 693 y(the)33 b(recurrence)i(relation)1672 904 y Fk(T)1729
558 919 y Fi(0)1769 904 y Fl(\()p Fk(x)p Fl(\))28 b(=)f(1)1669
559 1229 y Fk(T)1726 1244 y Fi(1)1766 1229 y Fl(\()p Fk(x)p
560 Fl(\))h(=)f Fk(x)1218 1554 y(T)1275 1569 y Fj(n)p Fi(+2)1413
561 1554 y Fl(\()p Fk(x)p Fl(\))h(=)f(2)p Fk(x)c Fh(\001)e
562 Fk(T)1908 1569 y Fj(n)p Fi(+1)2046 1554 y Fl(\()p Fk(x)p
563 Fl(\))h Fh(\000)h Fk(T)2356 1569 y Fj(n)2403 1554 y Fl(\()p
564 Fk(x)p Fl(\))257 1765 y(for)32 b Fk(n)c Fh(\025)g Fl(0.)257
565 1982 y Fg(Theorem:)404 2200 y Fk(T)461 2215 y Fj(n)508
566 2200 y Fl(\()p Fk(x)p Fl(\))33 b(satis\014es)g(the)g(di\013eren)m(tial)
567 d(equation)j(\()p Fk(x)2178 2163 y Fi(2)2239 2200 y Fh(\000)23
568 b Fl(1\))f Fh(\001)f Fk(T)2568 2140 y Ff(00)2554 2224
569 y Fj(n)2613 2200 y Fl(\()p Fk(x)p Fl(\))h(+)g Fk(x)g
570 Fh(\001)g Fk(T)3062 2140 y Ff(0)3048 2224 y Fj(n)3095
571 2200 y Fl(\()p Fk(x)p Fl(\))g Fh(\000)h Fk(n)3406 2163
572 y Fi(2)3468 2200 y Fh(\001)257 2320 y Fk(T)314 2335 y
573 Fj(n)362 2320 y Fl(\()p Fk(x)p Fl(\))k(=)h(0)k(for)g(all)f
574 Fk(n)d Fh(\025)g Fl(0.)257 2537 y Fg(Pro)s(of:)404 2755
575 y Fl(Let)45 b Fk(F)62 b Fl(:=)868 2688 y Fe(P)956 2715
576 y Fd(1)956 2779 y Fj(n)p Fi(=0)1110 2755 y Fk(T)1167
577 2770 y Fj(n)1214 2755 y Fl(\()p Fk(x)p Fl(\))p Fk(z)1394
578 2718 y Fj(n)1487 2755 y Fl(b)s(e)45 b(the)g(generating)g(function)f(of)
579 h(the)g(sequence)j(of)257 2875 y(p)s(olynomials.)41 b(The)33
580 b(recurrence)i(is)d(equiv)-5 b(alen)m(t)32 b(to)g(the)h(equation)1222
581 3085 y(\(1)22 b Fh(\000)g Fl(2)p Fk(x)h Fh(\001)e Fk(z)27
582 b Fl(+)22 b Fk(z)1825 3044 y Fi(2)1865 3085 y Fl(\))g
583 Fh(\001)g Fk(F)42 b Fl(=)27 b(1)22 b Fh(\000)h Fk(x)f
584 Fh(\001)g Fk(z)257 3303 y Fg(Pro)s(of)38 b(1:)404 3520
585 y Fk(F)j Fl(is)27 b(a)h(rational)d(function)i(in)g Fk(z)t
586 Fl(,)i Fk(F)42 b Fl(=)1925 3481 y Fi(1)p Fd(\000)p Fj(xz)p
587 1845 3497 326 4 v 1845 3555 a Fi(1)p Fd(\000)p Fi(2)p
588 Fj(xz)s Fi(+)p Fj(z)2137 3536 y Fc(2)2181 3520 y Fl(.)g(F)-8
589 b(rom)26 b(the)i(theory)g(of)f(recursions)257 3641 y(with)e(constan)m
590 (t)h(co)s(e\016cien)m(ts,)i(w)m(e)e(kno)m(w)g(that)f(w)m(e)h(ha)m(v)m
591 (e)h(to)e(p)s(erform)f(a)g(partial)f(fraction)257 3761
592 y(decomp)s(osition.)59 b(So)39 b(let)e Fk(p)p Fl(\()p
593 Fk(z)t Fl(\))h(=)g Fk(z)1618 3725 y Fi(2)1684 3761 y
594 Fh(\000)27 b Fl(2)p Fk(x)f Fh(\001)g Fk(z)31 b Fl(+)25
595 b(1)39 b(b)s(e)f(the)h(denominator)e(and)h Fk(\013)g
596 Fl(=)257 3881 y Fk(x)23 b Fl(+)433 3800 y Fh(p)p 516
597 3800 266 4 v 81 x Fk(x)571 3853 y Fi(2)633 3881 y Fh(\000)g
598 Fl(1)32 b(and)h Fk(\013)1067 3845 y Fd(\000)p Fi(1)1193
599 3881 y Fl(its)f(zero)s(es.)45 b(The)33 b(partial)d(fraction)i(decomp)s
600 (osition)f(reads)916 4131 y Fk(F)41 b Fl(=)1263 4063
601 y(1)22 b Fh(\000)g Fk(xz)p 1134 4107 534 4 v 1134 4199
602 a Fl(1)g Fh(\000)g Fl(2)p Fk(xz)27 b Fl(+)22 b Fk(z)1627
603 4170 y Fi(2)1705 4131 y Fl(=)1819 4063 y(1)p 1819 4107
604 49 4 v 1819 4199 a(2)1894 4009 y Fe(\022)2082 4063 y
605 Fl(1)p 1965 4107 283 4 v 1965 4199 a(1)g Fh(\000)h Fk(\013)q(z)2280
606 4131 y Fl(+)2553 4063 y(1)p 2388 4107 378 4 v 2388 4199
607 a(1)f Fh(\000)h Fk(\013)2622 4170 y Fd(\000)p Fi(1)2716
608 4199 y Fk(z)2776 4009 y Fe(\023)257 4388 y Fl(hence)38
609 b Fk(T)589 4403 y Fj(n)636 4388 y Fl(\()p Fk(x)p Fl(\))d(=)922
610 4348 y Fi(1)p 922 4364 36 4 v 922 4422 a(2)967 4388 y
611 Fl(\()p Fk(\013)1068 4351 y Fj(n)1139 4388 y Fl(+)25
612 b Fk(\013)1303 4351 y Fd(\000)p Fj(n)1404 4388 y Fl(\).)55
613 b(Note)36 b(that)g(the)h(\014eld)f Fk(Q)p Fl(\()p Fk(x)p
614 Fl(\)\()p Fk(\013)q Fl(\),)i(b)s(eing)d(a)h(\014nite)g(di-)257
615 4508 y(mensional)28 b(extension)j(\014eld)e(of)g Fk(Q)p
616 Fl(\()p Fk(x)p Fl(\))h(in)f(c)m(haracteristic)g(0,)h(has)g(a)g(unique)g
617 (deriv)-5 b(ation)257 4628 y(extending)745 4589 y Fj(d)p
618 725 4605 77 4 v 725 4663 a(dx)854 4628 y Fl(on)44 b Fk(Q)p
619 Fl(\()p Fk(x)p Fl(\).)76 b(W)-8 b(e)43 b(can)h(therefore)g(try)f(to)g
620 (construct)i(an)e(annihilating)257 4749 y(di\013eren)m(tial)28
621 b(op)s(erator)h(for)g Fk(T)1335 4764 y Fj(n)1382 4749
622 y Fl(\()p Fk(x)p Fl(\))h(b)m(y)g(com)m(bination)e(of)h(annihilating)d
623 (di\013eren)m(tial)i(op-)257 4869 y(erators)i(for)e Fk(\013)788
624 4833 y Fj(n)864 4869 y Fl(and)h Fk(\013)1113 4833 y Fd(\000)p
625 Fj(n)1215 4869 y Fl(.)42 b(In)29 b(fact,)h Fk(L)1685
626 4884 y Fi(1)1753 4869 y Fl(:=)d(\()p Fk(\013)16 b Fh(\000)f
627 Fk(x)p Fl(\))2215 4830 y Fj(d)p 2194 4846 V 2194 4904
628 a(dx)2296 4869 y Fh(\000)g Fk(n)30 b Fl(satis\014es)g
629 Fk(L)2898 4884 y Fi(1)2937 4869 y Fl([)p Fk(\013)3027
630 4833 y Fj(n)3074 4869 y Fl(])e(=)f(0,)j(and)257 4990
631 y Fk(L)323 5005 y Fi(2)391 4990 y Fl(:=)d(\()p Fk(\013)c
632 Fh(\000)e Fk(x)p Fl(\))866 4950 y Fj(d)p 845 4966 V 845
633 5024 a(dx)953 4990 y Fl(+)g Fk(n)33 b Fl(satis\014es)g
634 Fk(L)1566 5005 y Fi(2)1606 4990 y Fl([)p Fk(\013)1696
635 4953 y Fd(\000)p Fj(n)1797 4990 y Fl(])28 b(=)f(0.)44
636 b(A)32 b(common)f(m)m(ultiple)e(of)j Fk(L)3128 5005 y
637 Fi(1)3200 4990 y Fl(and)g Fk(L)3455 5005 y Fi(2)257 5110
638 y Fl(is)g(easily)g(found)h(b)m(y)g(solving)f(an)g(appropriate)g(system)
639 i(of)e(linear)f(equations:)404 5254 y Fk(L)d Fl(=)f(\()p
640 Fk(x)694 5218 y Fi(2)737 5254 y Fh(\000)s Fl(1\))921
641 5157 y Fe(\020)1001 5215 y Fj(d)p 981 5230 V 981 5288
642 a(dx)1067 5157 y Fe(\021)1117 5180 y Fi(2)1160 5254 y
643 Fl(+)s Fk(x)1324 5215 y Fj(d)p 1304 5230 V 1304 5288
644 a(dx)1394 5254 y Fh(\000)s Fk(n)1532 5218 y Fi(2)1600
645 5254 y Fl(=)1703 5157 y Fe(\020)1753 5254 y Fl(\()p Fk(\013)c
646 Fh(\000)f Fk(x)p Fl(\))2099 5215 y Fj(d)p 2078 5230 V
647 2078 5288 a(dx)2187 5254 y Fl(+)g Fk(n)2343 5157 y Fe(\021)2396
648 5254 y Fh(\001)s Fk(L)2493 5269 y Fi(1)2561 5254 y Fl(=)2664
649 5157 y Fe(\020)2714 5254 y Fl(\()p Fk(\013)g Fh(\000)h
650 Fk(x)p Fl(\))3059 5215 y Fj(d)p 3039 5230 V 3039 5288
651 a(dx)3148 5254 y Fh(\000)f Fk(n)3305 5157 y Fe(\021)3358
652 5254 y Fh(\001)s Fk(L)3455 5269 y Fi(2)404 5388 y Fl(It)32
653 b(follo)m(ws)g(that)g(b)s(oth)g Fk(L)p Fl([)p Fk(\013)1427
654 5352 y Fj(n)1474 5388 y Fl(])c(=)g(0)k(and)h Fk(L)p Fl([)p
655 Fk(\013)2060 5352 y Fd(\000)p Fj(n)2162 5388 y Fl(])27
656 b(=)h(0,)k(hence)i Fk(L)p Fl([)p Fk(T)2849 5403 y Fj(n)2897
657 5388 y Fl(\()p Fk(x)p Fl(\)])28 b(=)g(0.)1852 5637 y(1)p
660 2 1 bop 257 573 a Fg(Pro)s(of)38 b(2:)404 801 y Fl(Starting)g(from)g
661 (the)i(ab)s(o)m(v)m(e)h(equation,)g(w)m(e)f(compute)g(a)f(linear)f
662 (relation)g(for)h(the)257 922 y(partial)31 b(deriv)-5
663 b(ativ)m(es)32 b(of)g Fk(F)14 b Fl(.)44 b(W)-8 b(rite)31
664 b Fk(@)1637 937 y Fj(x)1710 922 y Fl(=)1843 882 y Fj(d)p
665 1823 898 77 4 v 1823 956 a(dx)1942 922 y Fl(and)h(\001)2212
666 937 y Fj(z)2280 922 y Fl(=)c Fk(z)2461 882 y Fj(d)p 2443
667 898 72 4 v 2443 956 a(dz)2525 922 y Fl(.)44 b(One)33
668 b(computes)1282 1080 y Fe(\020)1332 1176 y Fl(1)21 b
669 Fh(\000)i Fl(2)p Fk(xz)k Fl(+)22 b Fk(z)1825 1135 y Fi(2)1865
670 1080 y Fe(\021)1937 1176 y Fh(\001)g Fk(F)41 b Fl(=)28
671 b(1)22 b Fh(\000)g Fk(xz)1222 1449 y Fe(\020)1272 1545
672 y Fl(1)g Fh(\000)g Fl(2)p Fk(xz)27 b Fl(+)22 b Fk(z)1765
673 1504 y Fi(2)1806 1449 y Fe(\021)1855 1472 y Fi(2)1917
674 1545 y Fh(\001)f Fk(@)2017 1560 y Fj(x)2062 1545 y Fk(F)41
675 b Fl(=)28 b Fk(z)f Fh(\000)22 b Fk(z)2490 1504 y Fi(3)1153
676 1818 y Fe(\020)1203 1915 y Fl(1)g Fh(\000)g Fl(2)p Fk(xz)27
677 b Fl(+)22 b Fk(z)1696 1874 y Fi(2)1737 1818 y Fe(\021)1786
678 1841 y Fi(3)1848 1915 y Fh(\001)g Fk(@)1954 1874 y Fi(2)1949
679 1939 y Fj(x)1994 1915 y Fk(F)41 b Fl(=)28 b(4)p Fk(z)2300
680 1874 y Fi(2)2362 1915 y Fh(\000)23 b Fl(4)p Fk(z)2560
681 1874 y Fi(4)1025 2188 y Fe(\020)1074 2284 y Fl(1)f Fh(\000)h
682 Fl(2)p Fk(xz)k Fl(+)22 b Fk(z)1568 2243 y Fi(2)1608 2188
683 y Fe(\021)1658 2211 y Fi(2)1719 2284 y Fh(\001)g Fl(\001)1850
684 2299 y Fj(z)1890 2284 y Fk(F)42 b Fl(=)27 b Fk(xz)g Fh(\000)c
685 Fl(2)p Fk(z)2423 2243 y Fi(2)2485 2284 y Fl(+)f Fk(xz)2687
686 2243 y Fi(3)719 2557 y Fe(\020)769 2654 y Fl(1)f Fh(\000)i
687 Fl(2)p Fk(xz)k Fl(+)22 b Fk(z)1262 2613 y Fi(2)1302 2557
688 y Fe(\021)1352 2580 y Fi(3)1414 2654 y Fh(\001)f Fk(@)1514
689 2669 y Fj(x)1559 2654 y Fl(\001)1640 2669 y Fj(z)1680
690 2654 y Fk(F)41 b Fl(=)28 b Fk(z)f Fl(+)22 b(2)p Fk(xz)2211
691 2613 y Fi(2)2273 2654 y Fh(\000)h Fl(6)p Fk(z)2471 2613
692 y Fi(3)2533 2654 y Fl(+)f(2)p Fk(xz)2784 2613 y Fi(4)2846
693 2654 y Fl(+)g Fk(z)2993 2613 y Fi(5)554 2927 y Fe(\020)603
694 3023 y Fl(1)g Fh(\000)h Fl(2)p Fk(xz)k Fl(+)22 b Fk(z)1097
695 2982 y Fi(2)1137 2927 y Fe(\021)1187 2950 y Fi(3)1248
696 3023 y Fh(\001)g Fl(\001)1379 2982 y Fi(2)1379 3048 y
697 Fj(z)1419 3023 y Fk(F)41 b Fl(=)28 b Fk(xz)f Fl(+)22
698 b(\(2)p Fk(x)1994 2982 y Fi(2)2056 3023 y Fh(\000)h Fl(4\))p
699 Fk(z)2292 2982 y Fi(2)2354 3023 y Fh(\000)f Fl(\(2)p
700 Fk(x)2595 2982 y Fi(2)2657 3023 y Fh(\000)h Fl(4\))p
701 Fk(z)2893 2982 y Fi(4)2955 3023 y Fh(\000)f Fk(xz)3158
702 2982 y Fi(5)404 3256 y Fl(Solv)m(e)32 b(a)h(6)22 b Fh(\002)g
703 Fl(6)33 b(system)g(of)f(linear)f(equations)i(o)m(v)m(er)h
704 Fk(Q)p Fl(\()p Fk(x)p Fl(\))f(to)f(get)1124 3476 y Fk(x)22
705 b Fh(\001)g Fk(@)1302 3491 y Fj(x)1346 3476 y Fk(F)36
706 b Fl(+)22 b(\()p Fk(x)1636 3434 y Fi(2)1698 3476 y Fh(\000)h
707 Fl(1\))f Fh(\001)f Fk(@)2012 3434 y Fi(2)2007 3500 y
708 Fj(x)2053 3476 y Fk(F)36 b Fh(\000)22 b Fl(\001)2332
709 3434 y Fi(2)2332 3500 y Fj(z)2372 3476 y Fk(F)42 b Fl(=)27
710 b(0)404 3696 y(This)34 b(is)g(equiv)-5 b(alen)m(t)34
711 b(to)g(the)g(claimed)f(equation)h(\()p Fk(x)2336 3659
712 y Fi(2)2399 3696 y Fh(\000)23 b Fl(1\))g Fh(\001)g Fk(T)2731
713 3636 y Ff(00)2717 3720 y Fj(n)2776 3696 y Fl(\()p Fk(x)p
714 Fl(\))g(+)g Fk(x)h Fh(\001)f Fk(T)3230 3636 y Ff(0)3216
715 3720 y Fj(n)3263 3696 y Fl(\()p Fk(x)p Fl(\))h Fh(\000)257
716 3816 y Fk(n)315 3780 y Fi(2)377 3816 y Fh(\001)e Fk(T)484
717 3831 y Fj(n)531 3816 y Fl(\()p Fk(x)p Fl(\))28 b(=)g(0.)257
718 4149 y Fb(References)62 4368 y Fl(1)146 b(Bruno)46 b(Haible:)69
719 b(D-\014nite)45 b(p)s(o)m(w)m(er)i(series)g(in)e(sev)m(eral)i(v)-5
720 b(ariables.)82 b Fa(Diploma)46 b(thesis,)501 4488 y(University)35
721 b(of)g(Karlsruhe,)f(June)h(1989.)44 b Fl(Sections)33
722 b(2.12)f(and)h(2.15.)1852 5637 y(2)p eop
725 userdict /end-hook known{end-hook}if
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