Combinatorial Numbers and Associated Identities: Table 1: Stirling Numbers

Transcription

1 Combiatoial Numbes ad Associated Idetities: Table : Stilig Numbes Fom the seve upublished mauscipts of H. W. Gould Edited ad Compiled by Jocely Quaitace May 3, 200 Notatioal Covetios fo Table Thoughout this table, we assume ad ae oegative iteges. We let B deote the th Bell umbe. This is a faily stadad otatio fo the Bell umbes. Howeve, thee ae may otatios fo Stilig umbes of the fist ad secod ids. The followig table lists equivalet otatios fo Stilig umbes of the secod id. Notatio Autho Souce S(, Joh Rioda Combiatoial Idetities, 968 S(, L. Calitz umeous papes, maily that of 97 S { 2 (, } H. W. Gould vaious papes cica 956 Doald Kuth At of Compute Pogammig S [ G. Pólya Notes o Combiatoics,978 C w ( ] J. G. Hage Combiatioe mit Wiedeholuge, 89 C + Niels Nielse 906 S Chales Joda 939 S ( Kal Goldbeg ad Tomliso Fot Bueau of Stadads, 959! 0 Diffeeces of zeo actuaial wo ( Goldbeg, Leighto, Newma, Zucema 976! B, H. W. Gould pivate otatio of the seve oteboos Table : Equivalet otatios fo S(,, a Stilig umbe of the secod id

2 As implied by the last lie of Table, all of the idetities i this volume will use Gould s oigial otatio B,. The eade is uged to emembe that B,!S(,. The ext table lists equivalet otatios fo Stilig umbes of the fist id. Notatio Autho Souce s(, Joh Rioda Combiatoial Idetities, 968 S (, L. Calitz umeous papes, maily that of 97 ( S [ ( ], H. W. Gould vaious papes cica 956 ( Doald Kuth At of Compute Pogammig ( S G. Pólya Notes o Combiatoics,978 ( [C ( ] J. G. Hage Combiatioe ohe Wiedeholuge, 89 ( C Niels Nielse 906 S Chales Joda 939 S ( Kal Goldbeg Bueau of Stadads, 959 ( I ( Goldbeg, Leighto, Newma, Zucema 976!C H. W. Gould pivate otatio of the seve oteboos Table 2: Equivalet otatios fo s(,, a Stilig umbe of the secod id As implied by the last lie of Table 2, all of the idetities i this volume will use Gould s oigial otatio C. The eade is uged to emembe that C s(,!. 2 Stilig Numbes of the Secod Kid B,!S(, Rema 2. Thoughout this chapte, we assume ad ae oegative iteges. We assume x, y, ad z ae eal o complex umbes. We also let [x] deote the floo of x fo ay eal x. 2. Basis Defiitio fo B, ( x x B, ( Applicatios of Equatio (2. ( B, { 0, 2, (2.2 2

3 ( 2 B, { 0, 3, 2 (2.3 ( B, ( (2.4 ( x + ( B, ( x (2.5 ( B, ( (2 (2.6 ( 2 ( 2 2 B, ( 2 (2.7 B, ( ( B, ( Explicit Fomulas fo B, B, ( B, ( (, 0 0 (2.9 ( ( ( ( Applicatios of Equatio ( Evaluatios of h x hx ( h ( ( (x + h ( x h B, (2. x x0 B, (2.2 3

4 2.3.2 Evaluatio of ( ( x ( x ( ( x x ( ( Applicatios of Equatio (2.3 ( x ( ( x x B,,, (2.3 ( x (,, (2.4 ( ( ( x x 2 x 3 ( 2 ( x + ( x(x, 2 (2.5 2 ( ( ( x x 2 x 3 ( 3 ( x + 3( x(x 2 ( x 4 + ( x(x (x 2, 3 (2.6 3 ( ( ( x x 2 x 3 ( 4 ( x + 7( x(x 2 ( ( x 4 x 5 + 6( x(x (x 2 + ( x(x (x 2(x 3, 4 ( ( x + ( ( x + x + + B, (2.8 ( ( x + + x + (2.9 ( + ( + ( 2 + B, (2.20 ( ( (2.2 4

5 ( + ( + ( ( ( + ( ( ( + ( 2 + ( ( + 2( + 2, (2.22 ( ( + 2( + 2 ( ( + 3( + 2( + 3, 2 (2.23, 3 (2.24 ( ( (2.25 ( ( ( 2 + B, (2.26 Rema 2.2 The followig idetity is Poblem 455, P.482, of the Ameica Math. Mothly, 953. lim α ( α ( ( αγ( α, (2.27 whee α is eal umbe which is ot a oegative itege, ad is a positive itege. ( x ( (z + y Evaluatio of x 0! Applicatios of Equatio ( ( z y ( ( x x ( B, (2.28 x x e x!! B, ( x x! ( x! 0 (2.30 5

6 cos x! si x! e cos x (cos(si x e cos x (cos(si x B, cos x si(si x! B, si x + si(si x! B, B, si x! cos x! (2.3 ( Guet s Fomula Guet s Fomula Let S be a fuctio of x. Let d S dx deote the th deivative of S with espect to x. The, ( x d S dx 2.4. Applicatios of Guet s Fomula x B,! d S dx. (2.33 x B, x, x < (2.34 ( x + x B, ( x (2.35 ( + B, ( ( + ( B +,, (2.37 ( + ( 2 ( ( [ + 2 ] (2 + ( 4 (2.38 (2.39 6

7 ( 2 ( (2.40 ( 3 ( ( + 8 (2.4 ( 4 ( (2.42 ( + (( ( + [ 2 + ] + 2 B, ( ( x ( + x ( 2 ( x ( + x B,, x (2.44 ( B 2, (2.45 ( ( 3 2 3( ( ( 4 2 7( 6( ( ( ( 2 8 ( ( 2( 3 6 (2.46 (2.47 ( (2 + (x 2+ (2 +! ( x si x + π 2 B, (2.48 ( (2 (x 2 (2! ( x cos x + π 2 B, (2.49 7

8 2.5 Popeties of B, 2.5. Recuece Fomulas fo B, B, ( ( ( B, (2.50 B + +,+ (B ( + +,+ + B, Rema 2.3 Fo the defiitio of,, we efe the eade to Volume 2, Boo, Chapte 8, Equatio (8.. (2.5 ( + B + +,+ ( + [ ( + B +,+ ( B,] [( +, ] ( B, (2.52 a+ [ ] ( +, ( B, ( + B + +,+, (2.53 ( B, ( B, ( a B a,a,, a 0 (2.54 B, (!B +, +,, ( Covolutio Popety fo B, ( B, B, B +,+ (2.56 Applicatios of Equatio (2.56 ( ( (z + ( z B + +,+ (2.57 ( ( (z + 2z ( +! ( ( ( (z z 2 2z ( + 2! (

9 2.6 Expasios of ( e x x Rema 2.4 I this sectio, we assume D xf(x is the th deivative of f(x with espect to x. ( e x x B +, x ( +! (2.60 D x D x ( e x x0 x ( ( ( x + e x x0 (! ( +! B+, (2.6! ( + B +, (2.62 ( Polyomial Expasios fom B, Rema 2.5 Thoughout this sectio, we assume f(x is a polyomial of degee. f(x ( ( x ( f( ( Applicatios of Equatio (2.63 ( x + ( ( ( x + ( (2.64 ( mx ( ( x ( ( m m, whee m is a complex umbe (2.65 ( x ( x ( ( x ( ( ( x ( p[ + ] ( (2.66 ( (2.67 p 9

10 ( 3 3 ( ( ( 3 ( (2.68 p p[ +2 3 ] ( 2 2 ( ( ( 2 ( p p[ + 2 ] ( 2 ( Polyomial Seies via B, Rema 2.6 Thoughout this sectio, we assume f(x is a polyomial of degee. x f( ex! 2.8. Applicatios of Equatio (2.70 x! ( ( f( (2.70 Dobisi s Fomula ( + x! ex (! e ( x! (2.7 ( B, ( Bell Numbes B Rema 3. Thoughout this chapte, we assume ad ae oegative iteges. We assume x, y, ad z ae eal (complex umbes. We also let [x] deote the floo of x fo ay eal x. 3. Defiitio of B Stilig s Fomula B e B B,! (3.!, (3.2 0

11 3.2 Popeties of Bell Numbes 3.2. Bell Numbe Recueces ( B + B, B 0 (3.3 ( B ( B + ( Expoetial Geeatig Fuctio e ex 3.3 Schäfli s Modified Bell Numbe Recuece x! B (3.5 Rema 3.2 The idetities of this sectio ae foud i L. Schläfli s O a geealizatio give by Laplace of Lagage s Theoem, Quately Joual of Pue ad Applied Mathematics, Vol. 2 (858, pp The eade is also efeed to Olive A. Goss s, Pefeetial Aagemets, Ameica Math. Mothly, Vol. 69 (962, pp Schläfli s Recuece ( A A,, A 0 ( Alteative Foms of Equatio (3.6 2A ( A, (3.7 A B, (3.8

12 3.3.2 Expoetial Geeatig Fuctio Let A (x B, x. The, x(e t A (x t!. (3.9 2 e t A t! (3.0 0 t! ( x B, (e t ( (e t x + ( Dobisi Numbes D 3.4. Defiitio of D D ( B,! (3.2 Dobisi s Fomula D e (! ( Popeties of Dobisi Numbes Recuece Relatio ( + D ( ( D +, D 0 (3.4 Expoetial Geeatig Fuctio e ex x! D (3.5 2

13 3.5 Fuctioal Bell Numbe Recuece Rema 3.3 I this sectio, we assume f is a eal o complex valued futio ove the oegative iteges. We also assume p is a positive itege. p ( + p ( f( ( + f( (p + ( + p + f(p (3.6 4 Shifted Stilig Numbes of the Secod Kid A, Rema 4. Thoughout this chapte, we assume ad ae oegative iteges. We assume x, y, ad z ae eal o complex umbes. We also let [x] deote the floo of x fo ay eal x. 4. Defiitio of A, x ( x + A, ( Explicit Fomulas fo A, A, A, ( + ( ( (4.2 ( + ( + ( Relatioships Betwee A, ad B, B, ( A,, 0 (4.4 A, ( ( + 0 ( ( + B, (4.5 A, ( ( + B, (4.6 3

14 4.3. Applicatios of Equatio (4.4 A,! (4.7 Rema 4.2 The followig two idetities ae foud i Robet Stalley s A Geealizatio of the Geometic Seies, Ameica Math. Mothly, May 949, Vol. 56, No. 5, pp x x ( x + A,, x <, (4.8 x ( + ( + ( + x, x <, (4.9 ( x + Rema 4.3 The followig idetity ca be foud i T. M. Apostol s O the Lech Zeta Fuctio, Pacific Joual of Mathematics, Vol., No. 2, Jue 95, pp z ye z ( z +!(y + y A,, y <, ye z < ( Popeties of A, 4.4. Idice Equivalece Popety A, A +, { 0, (, 0, 0, ( Recuece Relatio Guet s Fomula A,+ ( + 2 A, + A,, (4.2 Let S be a eal o complex valued futio ove the set of oegative iteges. Let d S deote the dx th deivative of S with espect to x. The, ( x d ( x d S S A, (4.3 dx! dx 4

15 4.4.4 Evaluatio of ( + A,, 0, ( Wopitzy/Nielse Numbes B,q Rema 5. Thoughout this chapte, we assume, q, m, ad ae oegative iteges. We assume x, y, ad z ae eal o complex umbes. We also let [x] deote the floo of x fo ay eal x. 5. Defiitio of B,q ( q B,q ( ( ( Coectio to A, A, B,+ ( + ( ( ( Popeties of B,q 5.2. Idex Shift Popety { B,m+ + ( m+ B +,m+ 0, m ( ( m+, m 0 ( Relatioships to B, ( m+ B, B, m+ m+ ( B m,m+, m (5.4 ( m B m,m+, m (5.5 5

16 B,m+ ( Applicatios of Equatio (5.4 m ( m ( B m,, m (5.6 Rema 5.2 The followig idetity is a fomula give by N. Nielse i Taité élémetaie des ombes de Beoulli, Pais, 924, pp Nielse s Fomula Recuece Relatio ( m+ x Guet s Fomula m+ ( x + m B,m+, m (5.7 B +,m+ (m + B,m + B,m, m + (5.8 Let S be a eal o complex valued futio ove the set of oegative iteges. Let d S deote the dx th deivative of S with espect to x. The, ( x d m+ S ( m+ dx B,m+ 5.3 Polyomial Expasios fom Nielse s Fomula ( x d S m! dx, m (5.9 Rema 5.3 Thoughout this sectio, we assume f(x is a abitay polyomial of degee m, amely, f(x m a x. Nielse s Polyomial Expasio f(x ( m m 5.3. Applicatios of Equatio (5.0 ( ( x + m + ( f( (5.0 m Rema 5.4 Thoughout this subsectio, we assume p is a positive itege. 6

17 ( p x m ( ( ( p x + m + + ( m+p (, p m (5. m ( p x p + ( x + p ( x + m ( ( x + m + ( m ( m ( ( p p + + (,, p (5.2 ( +, m (5.3 ( x ( x + ( ( ( + + ( (5.4 ( px m ( x + ( m m ( ( m + ( p p, m (5.5 ( x ( ( ( x + + ( ( (5.6 A th Diffeece Applicatio of Equatio (5.0 Assume f(x is a polyomial of degee m. The, ( ( f( ( m+ Applicatios of Equatio (5.7 ( ( p ( ( p m ( ( m + ( f(. (5.7 m ( ( p + + (, p (5.8 ( ( + ( { p + 0, p < (p!,! p p (5.9 7

18 2 ( ( ( ( 3 2 ( 2 ( ( ( 3 + ( 3 3 (3 + ( ( 3 2 ( 2 (5.2 2 ( ( 2 3 ( 3 2 ( 2 ( 3 2 ( ( ( ( 2 ( ( ( ( 2 ( Geealized Nielse Expasios Rema 5.5 Thoughout this sectio, we assume f(x is a abitay polyomial of degee, amely, f(x a x. Nielse s Fist Geealized Polyomial Expasio: Assume m. The, m+ f(x + y ( m ( ( x + m + ( f( + y. (5.24 m Nielse s Secod Geealized Polyomial Expasio f(x + y ( ( x ( f( + y ( Applicatios of Equatio (5.24 ad (5.25 Rema 5.6 I this subsectio, we assume p is a positive itege. f(x ( m m ( m + ( f( m + x, m (5.26 8

19 ( p m+ x + y ( ( ( p x + m + + y ( m ( (5.27 m ( p m+ x + y ( x + ( p+m m ( ( p m + y + (, p m (5.28 ( p x + y p+ ( x + p ( ( p + ( p y + (5.29 ( p x y p+ ( x + p ( ( p + ( p + y + (5.30 ( p y p ( ( p p + p + y + ( (5.3 p ( ( p p + p + ( 0, (5.32 ( ( p p + p ( (5.33 p + ( ( ( (5.34 ( ( ( (5.35 9

20 ( ( 2 ( 2 + (2 + ( 2 ( ( ( ( ( ( ( ( (5.38 ( px + py ( ( ( x p p + py ( (5.39 ( m+ px + py ( ( x + m + ( m ( m ( p p + py, m (5.40 ( m+ px + py ( x + ( m+ m ( ( m + p p py + (, m (5.4 ( 2x + 2y + ( ( ( x y + ( (5.42 ( 2y + 2 ( ( y ( (5.43 ( ( ( {, 0 0, (

21 ( ( ( ( 2 ( ( ( y y + ( (5.45 (5.46 ( ( ( + (5.47 Rema 5.7 The followig idetity is the solutio of B. C. Wog s Poblem 3399 of The Ameica Math. Mothly, Vol. 36, No. 0, Decembe 929. [ 2 ] ( ( ( + (5.48 ( ( ( (5.49 Rema 5.8 The followig idetity is the solutio of B. C. Wog s Poblem 3426 of The Ameica Math. Mothly, May 930. [ 2 ] ( ( (, (5.50 [ 2 ] ( ( ( ( ( ( ( + 2 {, , ( ( ( + p p + + p ( (5.5 (5.52 (5.53 2

22 [ (p p ] ( ( + ( ( p p + p (5.54 [ 2 3 ] ( ( ( ( ( Nielse Numbes βp m, (α Rema 5.9 Thoughout this sectio, we assume p is a oegative itege while α is a eal o complex umbe Defiitio of βp m, (α β m, p (α p ( m + ( (α + p (5.56 Rema 5.0 This defiitio is foud o Page 3 of Taité élémetaie des ombes de Beoulli, by Niels Nielse, Gauthie-Villas, Pais, Relatioship to Wopitzy Numbes β m, p (α β m, p (0 B p,m+ ( Polyomial Expasios via Nielse Numbes ( α Bp,m+ (5.58 Rema 5. This idetity is foud o Page 28 of Taité élémetaie des ombes de Beoulli by Niels Nielse. ( m+ (x α m+ ( x + m β m, (α, m (

23 6 Stilig Numbes of the Fist Kid C s(,! Rema 6. Thoughout this chapte, we assume, q, m, ad ae oegative iteges. We assume x, y, ad z ae eal o complex umbes. We also let [x] deote the floo of x fo ay eal x. 6. Basis Defiitio of C 6.. Deivatives of ( x ( x C x (6. Rema 6.2 Fo this subsectio, we let D p x deote the p th deivative with espect to x. D 2 x ( x D x D x ( x ( x + x, (6.2 ( x x0 ( C, (6.3 ( ( x 2 + x, (6.4 ( + x 2 D 2 x ( x x0 2 ( 2C 2, 2 ( Popeties of C 6.2. Recuece Fomula ( + C + C C, ( Covolutio Fomula 0 C C ( + C+ (6.7 23

24 6.2.3 Othogoality Popeties ( 0 B, C ( 0 C B, Rema 6.3 The followig idetity is a fomula of Fa R. Olso fom The Ameica Math. Mothly, Octobe 956, Vol. 63, No. 8, p. 62. { C B + 0, <, (6.0, Ivesio Fomulas Assume b is idepedet of, ad a B,b. The, b Let. Assume b is idepedet of, ad a Alteatig Sum Fomulas ( C ( (6.8 (6.9 C a. (6. B,b. The, b Ca. (6.2 ( ( C (6.3 ( C, ( Fuctioal Expasios Ivolvig C 6.3. Expasios of ( + x ( + x! ( C x, (6.5! ( x B, + x,, x < (6.6 24

25 6.3.2 Expasios of ( x+ ( x + x ( C q x (6.7 ( x + If q x, the, C ( ( + q. ( Expasios of ( x Ivolvig B x, Rema 6.4 Fo this subsectio, we defie, fo abitay eal o complex x, the geealized Stilig umbe of the secod id B x, as follows. B x, ( ( + x (6.9 Note that Equatio (6.9 implies z x B x, ( z. (6.20 ( x B x, ( C + C (6.2 x + C 2 ( ( Bx, (6.22 ( C + C (6.23 C ( (, 2 (

26 Two Othogoality Relatioships Let B x, ( x E, whee E ( ( + (. ( The, ad ( ( C + C E 0, (6.26 E ( ( C 0 + C. ( Factioal Biomial Sum Expasios ( ( x ( x+ ( ( x+ [ 2 ] C + 2+ x2 (6.28 ( 2 ( 3 ( 4 ( ( x ( x ( x ( x ( x+ ( x+ ( x+ ( x+ (x + x (x + 2(x + 6x (x + 3(x + 2(x + x x (x + 4(x + 3(x + 2(x + (6.29 (6.30 (6.3 (

27 6.3.5 Expasio of ( + z z Rema 6.5 The followig idetity is foud i O the Expasio of (+x x i Ascedig Powes of x, by Pecival Fost, Quately Joual of Mathematics, Lodo: Vol. 7, No. 28, Feb ( + z z z C (! 0 C +, whee (6.33 ( ( + ( +! ( ( s s + (6.34 s s0 6.4 A Deivative Expasio Ivolvig C Rema 6.6 Thoughout this subsectio, we let D xf(x deote the th deivative of f(x with espect to x.! D xf(l x x 6.4. Applicatios of Equatio (6.35 C D xf(z, whee z l x (6.35 D z (l z!! z! (l z C (6.36 D z (l z 2 2( (! z ( l z, 2 ( Ivesio of Equatio (6.35 B, z! D z (l z! (6.38 C ( z!! ( ( (l z Dz (l z (

28 Applicatios of Equatio (6.39 C!! D z (l z z (6.40 (l(z +! C z, z < (6.4 (z + x x C z (6.42 (x + x x [ 2 ] C (6.43 x! (e x C, e x < ( Explicit Fomulas fo C usig Beoulli Polyomials ad B, Rema 6.7 Let t be a eal (complex umbe. We defie B ( (t as the geeal Beoulli polyomial of ode ad degee by the geeatig fuctio elatio B ( (tx! x e tx, x < 2π. (6.45 (e x A excellet efeece fo popeties of B ( (t is Calculus of Fiite Diffeeces by Chales Joda, Secod Editio, Chelsea Publishig, New Yo, 947. C!(! B(+ ( (6.46 ( ( + C x!(! D x e x e x x0 (

29 Ivese of Idetity (6.49 C C C C!(! ( ( +!C ( + (!!! B+, (!! (!! C ( + ( ( + x Dx e x x0 (6.48 ( ( + +! ( +! B+, (6.49 (! ( C (6.50 ( ( + ( s! s + ( + s + s! B+s+s s+s,s+s (6.5 s0 ( ( s + s + s0 (!! s0 ( s (s + s! ( +s+s ( ( + + ( + ( ( s s!( + + s ( +s B +s ( ( ( ( s s + s + s0 B +s+s s+s,s+s (6.52 s,s (6.53 s! ( +s s B +s s,s (

30 6.6 Schläfli s Fomulas fo C Rema 6.8 The idetities of this sectio ca be foud i the followig two papes, both witte by L. Schläfli: Su les coefficiets du développemet du poduit ( + x( + 2x...( + ( x suivat les puissaces ascedates de x, Jou. eie u. agew. Math., Volume 43, 852, pp. -22: Egäzug de Abhadlug übe die Etwicelug des Puduts ( + x( + 2x...( + ( x (x, Jou. eie u. agew. Math., Volume 67, 867, pp C (! ( + ( +! B+, (6.55 C! ( ( + ( + +! B+, ( Ivese Relatios of Equatio (6.55 B, ( (! ( + ( + ( +!C + (

31 7 Appedix A: Cotou Itegal Fomulas fo Stilig Numbes Rema 7. Thoughout this appedix, we assume i. We also let γ deote a simple closed cuve aoud the oigi, amely, the uit cicle. 7. Cotou Itegals fo B, B, B,! (e z dz (7. 2πi γ z + (!(! e z (e z dz (7.2 2πi γ z 7.2 Cotou Itegals fo C C 2πi γ C 2π 2π 0 ( z dz z+ (7.3 ( e it e it dt (7.4!C (l(z + dz (7.5 2πi γ z +!C 2πi γ!c 2πi γ (!!C 2πi z e z dz (7.6 (e z + z dz (7.7 (e z γ z dz (7.8 (e z 3

32 7.2. Extesio of Idetity (7.7 Rema 7.2 I the followig idetity, assume α is a complex umbe with R(α >. Γ(α 0 z α (e z dz C α+ (7.9 8 Appedix B: Asymptotic Appoximatios fo Stilig Numbes Rema 8. May of the idetities i this appedix ae foud i Chales Joda s O Stilig s umbes, Tohou Math. Joual, Vol. 37, 933, pp Appoximatios Ivolvig B, B +, lim B, (8. lim B +, ( + 2!2 ( Appoximatios Ivolvig C C + < e+,, 0 (8.3!! C lim (8.4 2!2 lim C+, whee γ is Eule s costat (8.5 (l + γc Rema 8.2 The followig idetity, due to H. W. Bece, is foud i the Ameica Math. Mothly, Vol. 50, 943, Page 327. lim, ( +!C + ( + +!C ++ e (8.6 32

33 9 Appedix C: Numbe Theoetic Defiitios of Stilig Numbes 9. Kamp-Ettighause Defiitios of Stilig Numbes S (, ( ( 2...( γ + α!α 2!α 3!...2 α 3 α 24 α 3, (9. whee the summatio is ove all possible iteges α β such that α + 2α 2 + 3α , ad γ α + α 2 + α S 2 (, ( + ( +...( + γ + α!α 2!α 3!...(2! α (3! α 2(4! α 3, (9.2 whee the summatio is ove all possible iteges α β such that α + 2α 2 + 3α , ad γ α + α 2 + α Iteated Defiitios of S (, ad S 2 (, S (, (9.3 S (, i+ i i i+ + i, with + 0 (9.4 S 2 (, (9.5 S 2 (, i i i+ i, with + ( Restatemets of Equatios (9.3 ad (9.5 S (, , whee i is a positive itege (9.7 S 2 (, < 2 < 3 <...< , whee i is a positive itege (9.8 33

34 9.2.2 Applicatio of Equatio (9.5 S (,! i i+ i i 9.3 Popeties of S (, ad S 2 (, i, with + + (9.9 S (, (!C (9.0 S 2 (,! B+, (9. S (, S (, + S (,,, (9.2 S 2 (, S 2 (, + S 2 (,,, (9.3 S (, 9.3. Geealizatio of S 2 (, ( + ( + S 2 (, (9.4 S 2 (, S (, (9.5 S (, S 2 (, (9.6 Rema 9. I the followig idetity, we assume z is a abitay eal o complex umbe. S 2 (z, ( ( + z z S 2 (, ( Basis Expasios Ivolvig S (, ad S 2 (, Rema 9.2 I the followig two fomulas, we assume x is a eal o complex umbe. ( + x S (, x, (9.8 ( x S 2 (, x,, x < (9.9 34

35 9.3.3 Hage Recueces Rema 9.3 The followig idetities ae foud i Hage s Syopsis de hoehee Mathemati, Beli 89, Volume I, Page 60. S (, ( S (,, ( C ( S 2 (, ( ( C (9.2 + ( S 2 (,, ( B +, ( B, + (