Combinatorial Identities: Table I: Intermediate Techniques for Summing Finite Series

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1 Combiatorial Idetities: Table I: Itermediate Techiques for Summig Fiite Series From the seve upublished mauscripts of H. W. Gould Edited ad Compiled by Jocely Quaitace May 3, 00 Coefficiet Compariso Remar. Throughout this chapter, we will assume, uless otherwise specified, that x is a arbitrary complex umber ad is a oegative iteger.. Expasios of ( + x (a + x Remar. I this sectio, we will assume a is a arbitrary complex umber. ( ( ( (a a (... Equatio (. with a 0 ( ( ( (. ( ( + ( ( (.3 ( ( + ( ( (.4 ( ( + (, (.5

2 .. Equatio (. with a Special Case of the Vadermode Covolutio ( ( (.6..3 Equatio (. with a ( ( ( 4 ( ( (.7..4 Equatio (. with a Remar.3 The followig Idetity, due to Leo Moser, is Problem E799 of The America Math. Mothly, 948, P. 30. ( (..5 Shifted Versio of Equatio (. + ( ( (a + Equatio (.9 with a ( ( + ( (.8 ( ( a +, (.9 + (, (.0 + Equatio (.9 with a 0 + ( ( ( 0 (. +

3 . Expasios of (αx + β p ( + x q Remar.4 I this subsectio, we assume α ad β are arbitrary real or complex umbers. We ote that q is a arbitrary real umber, while p is a oegative iteger. Remar.5 The followig idetity is foud i Wilhelm Luggre s Et elemært bevis for e formel av A.C. Dixo, Nors Matematis Tidssrift, 9.Ågag, 947, pp p ( ( p q α p β.. Dixo Sums p ( ( p q + (α β p β (. Dixo s Idetity ( ( 3 ( ( 3 ( (.3 ( ( 3 ( ( ( 4 + ( (.4 ( 3 ( ( ( ( + ( (.5.3 Expasios of ( + x ( x ( ( ( ( ( ( 4 + ( ( (.6 (.7 ( + ( 4 ( (, (.8 3

4 .4 Expasios of ( x ( + x Remar.6 I this sectio, we assume r is a oegative iteger. r ( ( { 0, if r is odd ( ( r ( r (.9 r, if r is eve.4. Restatemets of Equatio (.9 First Restatemet of Equatio (.9 r+ ( ( ( 0 (.0 r + r ( ( ( ( ( r r r (. Secod Restatemet of Equatio (.9 r ( ( ( ( ( [ r + ( ] r [ r r, (. ] where, for ay real umber x, we let [x] deote the greatest iteger i x..4. Applicatios of Equatio (. r ( ( r ( ( r ( r( r (.3 ( r r ( ( ( r r + ( r ( r (.4 r ( ( + r ( r ( r (, r (.5 r 4

5 r ( ( ( ( ( ( r + r r ( r (.6 r ( ( r ( ( ( r ( r + r ( r ( ( (.7 r ( ( ( ( ( + ( (.8 [ ] ( 4 ( 4 + ( 4 ( + + ( 4 ( (.9 [ ] ( + 4 ( ( 4 ( 4 ( ( ( ( ( 4 ( 4 + ( Applicatios of Equatio (.9 to Expasios of ( + x (x (, (.30 ( 4 (.3 ( 4 + (.3 + ( ( ( 0, if is odd (.33 ( ( ( ( ( 4 ( ( ( ( ( ( [ ] ( + ( [ ] (.34 (.35 5

6 ( ( + ( 0, if is odd (.36 ( ( ( +.5 Expasios of ( x m ( + x m ( ( (.37 ( ( x x ( ( x ( ( ( (.38 ( ( [ x x ] ( ( x x ( ( ( ( ( x x x ( ( (.39 (.40 ( ( x x ( ( + x, (.4! ( ( x x (.5. Applicatios of Equatio (.40 ( ( x (.4 [ ] ( ( x x (( + ( ( x (.43 [ ] ( ( x x + (( ( ( x, (.44 6

7 .5. Special Cases of Equatio (.40 ( ( ( ( ( ( [ + ( ] [ ] ( ( ( 4 ( (.45 (.46 ( ( ( 4 ( +, (.47 (! ( ( ( ( ( 3 ( ( ( ( + ( ( 3 ( + ( 3 ( 4 (.48 (.49 ( ( ( + ( ( + ( ( ( ( (.50 ( + Remar.7 I the followig idetity, we evaluate ( (! by the Gamma fuctio, amely, (! Γ 3. ( ( ( + π ( + ( 4+ ( ( ( +! ( ( ( ( + (.5 ( + 3, (.5 ( +( ( + + ( (4 + ( 4! ( + 3, (.53 7

8 Remar.8 The followig two idetities are foud i C. va Ebbehorst Tegberge, Über die Idetitäte...etc. Nieuw. Arch. Wisde., Vol. 8 (943, pp. -7. We assume a is a oegative iteger. ( ( ( ( a ( ( a ( a a, a 0 (.54 a a a ( ( a ( + a ( +a ( +a +a a ( a a, (.55.6 Expasios of ( + x z ( x z Remar.9 I the followig sectio, we assume z is a arbitrary complex umber. Mitlag-Leffler s Polyomials ( ( z ( z ( z +.7 Expasios of ( + z p x ( + z y px ( +, (.56 Remar.0 I the followig sectio, we assume z ad y are arbitrary complex umbers, while p is a positive iteger. A referece for these expasios is Aother Note o the Hermite Polyomials by M. P. Drazi, America Math. Mothly, Vol. 64, 957, pp [ ] ( ( x y x ( ( x y ( (.57 [ ] ( ( x x ( ( x 3x ( (.58 [ 3 ] ( ( x y 3x 3 ( ( x y ( 3 (.59 8

9 ( ( ( x 3x ( 3 ( x 3 ( x 3 ( [ 3 ] + ( [ + 3 ] ( [ 3 ] ( [ 3 ] (.60 [ 3 ] ( ( x x 3.8 Expasios of (x + x ( ( x 4x ( 3 (.6 Remar. I the followig sectio, we will assume is a oegative iteger, ad y is a arbitrary complex umber. ( ( (.8. Applicatios of Equatio (.6 ( ( (.6 ( ( ( ( (.63 ( ( ( (.64 [ ] ( ( ( (.65 ( ( ( (.66 9

10 .8. Geeralizatios of Equatio (.64 ( ( ( x x (.67 [ ] ( ( ( x x x (.68 [ ] ( ( x x ( x (.69 ( ( x y + ( ( x y (.70 ( ( x y ( ( x x + y (.7 ( ( x y + ( x ( x + y (.7 Remar. The followig two idetities are associated with Problem 3803 [936, P.580; 938, pp ] i The America Math. Mothly ( ( ( x x x ( ( ( ( x 4x ( ( + ( ( x (.73 (.74 0

11 .9 Expasios by I. J. Schwatt Remar.3 I the followig sectio, we assume z is a arbitrary complex umber. We also let ((z deote the coefficiet of z i the power series expasio. For a referece, see Itroductio to the Operatios with Series by I. J. Schwatt, Uiv. of Pa. Press, 94, pp A reprit of this boo was issued i 96 by Chelsea Publ. Co., New Yor, N. Y. [ ] [ ] [ ] ( ( x x ((z ( + z x (.75 ( ( x x x ((z (z + z + x (.76 ( ( x x 3 ((z (z + 3z + x (.77.0 Coefficiet Comparisos Ivolvig the Compaio Biomial Theorem Remar.4 I this sectio, we will assume, uless otherwise specified, that a ad r are arbitrary real or complex umbers. ( ( ( a + r + a + r + + (.78 [ ] ( ( ( r + r + r + + ( ( r + r + ( r ( 4 ( ( r + r + ( r + + (.79 ( [ r + ] [ ], (.80 ( r + (.8 ( ( r + r ( r (.8 ( ( + i ( i (.83 i i0

12 Liear Algebra Techiques Applied to Series Remar. Throught this chapter, we assume, uless otherwise specified, that is a oegative iteger. We let x ad z deote arbitrary real or complex umbers. If x is a real umber, we let [x] deote the iteger part of x.. Series Derived from M Remar. Throughout this sectio, we assume M is a matrix of complex etries with two distict eigevalues, t ad t. [ ] i0.. Applicatios of Equatio (. i0 ( i (t t i (t + t i ( i i [ ] ( t i ( i ( i ( + t i t t t t (. t + (t ( + t, t (. [ ] ( i z i (z z + + ( (z + i0 i (z z (4z z (.3 [ ] i0 ( i ( i i i + (.4 ( + ( ( ( + (.5 [ ] ( i i + + ( i 3 i0 (.6 ( + ( (.7

13 [ ] ( i ( ( 5 + i + F, (.8 5 i0 where F is the th Fiboacci umber give by the recurrece F F + F, F 0, F. [ ] ( ( ( +i ( +i 3 ( 3+i 3, where i (.9 [ ] ( ( Limitig Case of Equatio (. i0 i0 ( [ 3 ] + ( [ + [ ] i0 ( i ( i i 3 ], 3m + a, (a 0,,,... (.0, (. i ( + i ( i (t t i (t + t i ( t+ t + (. i t t ( + (t t (t t t+ + t + (.3 t + t ( + i + ( i (t t i (t + t i+ ( t+ t + (.4 i + t t i0 ( + i ( i (t t i (t + t i+ ( t t (t t (.5 i + t t i0 ( + (t t + (t t t t (.6 + t + t (( + i + ( i (t t i (t + t i+ i + + ( + i ( (t + + t + (.7 i + 3

14 Remar.3 A referece for Equatio (.8 is the Solutio to Problem 4356 [949, 479] i America Math. Mothly, April 95, p. 68. i0 Limitig Case of Equatio (.8 ( + i ( i (t t i (t + t i+ i i + t+ + t + ( + (.8 i0 ( i ( + i i i i + ( + (.9 i0 ( + i ( i (t t i (t + t i ( t t (.0 i t t. Covolutios ad Determiats Remar.4 For Equatio (., we give the followig system of equatios a b c, (. where is a oegative iteger, ad b is idepedet of. We further assume that the a ad b are ow sequeces (with b 0 0. Our goal is to solve for c. This is doe i Equatio (.. Note the right side of Equatio (. ivolves the determiat of a matrix. c b + 0 a b b b 3... b a b 0 b b... b a 0 b 0 b... b a b 0... b a b 0 (. 4

15 Remar.5 For Equatio (.4, we are give the followig system of equatios a b c, (.3 where is a oegative iteger, ad b depeds o. Please ote that the superscript does ot deote a power of b, but oly the depedece o as well as. We further assume that a ad b are ow (with b 0 0. Our goal is to solve for c. This is doe i Equatio (.4. Note the right side of Equatio (.4 ivolves the determiat of a matrix. c i0 bi 0.. Applicatio of Equatio (.4 a b b b 3... b b 0 b b... b a a 0 b 0 b... b a b b a b 0 0 (.4 ( ( a ( a ( ( (... ( 3 a ( ( ( 0... ( a 0 ( ( 0... ( a ( a b 0 (.5 5

16 .3 Applicatio of Equatio (.4 to Reciprocal Expasios Remar.6 For Equatio (.7, we are give a power series represetatio for f(x, amely, f(x i0 a ix i, with a 0 0. We assume α is a iteger. Our goal is to calculate (f(x b α α x, (.6 where the superscripts o the b s are ot expoets, but ust a idicator of the etry i a array of umbers (See Remar.5. These calculatios are doe i Equatio (.7. Note the right side of Equatio (.7 is the determiat of a matrix. ad if, b α a +α 0! b α 0, where the α subscript o a a α 0 is a expoet, ( (α + a (α + a.... αa 0 ( a 0 (α + a.... α( a 0 0 ( a α( a ( 3a 0... α( 3a A Alterative Form of Equatio (.7 b α ( a +α 0! D, where D is the determiat of the followig ( ( matrix αa a αa (α + a a D 3αa 3 (α + a (α + a 3a αa (( α + a (( α + a (( 3α + 3a 3... (α + a 6

17 .4 Geeralized Cauchy Covolutio Remar.7 For Equatio (.9, we are give a power series i0 a ix i. Our goal is to calculate product of this power series, amely ( α a i x i Bx α, (.8 i0 where α is a iteger, a 0 0 if α is a egative iteger, ad the superscript o the B is ot a expoet (see Remar.5. Equatio (.9 provides a solutio for B. α Note B0 α a α 0, while if, B α ( α a α 0 f (, (.9 where the superscript o a 0 deotes a power, ad f (.5 Iverse Fuctio Expasios P i i i 0 a a a 3...a. (.30 Remar.8 Suppose we are give a power series y i a ix i, with a 0. Our goal is to ivert this series, amely to fid x α b αy α, where b f ( a, ad for, f ( f ( b ( f (3 f (3 f 3 ( a (+ f (4 f (4 f 3 (4 f 4 (4... 0, ( f ( f ( f 3 ( f 4 (... f ( with f ( defied by Equatio (.30. 7

18 3 Aihilatio Coefficiets Remar 3. I this chapter, we wor with the series c (x +, (3. where x is ay real or complex umber for which Equatio (3. is defied. Our goal is to determie the c such that all terms i the sum, except x ad the costat term, vaish simultaeously. All the formulas i this chapter are derived by this aihilatio procedure. Also, we assume is a positive iteger, ad m is a oegative iteger. ( (x r r (r + r (x + (3. r r0 ( ( r (x + r r (r + r (x (3.3 r r0 r0 ( (r + r ( r r r (3.4 m r0 ( r ( r m (r + r (x r r m ( (x + m (3.5 m ( ( + ( + (3.6 8

19 4 Kummer Series Trasformatio Remar 4. I this chapter, we start with the series a f(, where both ad a are oegative itegers. For a arbitrary fuctio ϕ(, we have Suppose that f( a i. lim f( ϕ( K 0 ii. S a f( coverges a iii. C a ϕ( coverges. The, Equatio (4. implies S ( f(ϕ( f( + f( ϕ(f( ϕ( a ϕ( (4. a ( K ϕ( f( + KC. (4. f( Equatio (4. is ow as the Kummer Series Trasformatio. A referece for this material is Korad Kopp s Theorie ud Awedug der uedliche Reihe Fourth Editio, Berli, 947. See Chapter VIII, P Applicatios of the Kummer Series Trasformatio π ( + ( + ( + 3 (4.3 ( +r r + r! (! ( + r!, r (4.4 Remar 4. I the followig idetity, we assume r is a positive iteger. r, r (4.5 r (r +! r! (r +! r + (r r! (4.6 9

20 4.. Kopp Applicatios of the Kummer Series Trasformatio Remar 4.3 I sectio, we assume p is a positive iteger, ad α is arbitrary real or complex umber, excludig the set of opositive itegers. Defie The, S p (α (. (4.7 ( + α( + α +...( + α + p S p (α (α + 3 p (p (α(α +...(α + p + p 3 (p S p+(α (4.8 S + ( ( 3 S 3! (4.9 S + ( π 6! ( 3! ( ( (4.0 ( + Limitig Case of Equatio (4. π 6 ( 3 ( π 8 ( ( (4. (4. 0

21 5 Series From Logarithmic Differetiatio Remar 5. Throughout this chapter, we will assume x is a arbitrary real or complex umber, while is a oegative iteger. 5. Basic Logarithmic Differetiatio Formula Assume a is a oegative iteger. If F (x ( + u (x, (5. a the F (x F (x a u (x + u (x, (5. where F (x is the derivative of F (x with respect to x. 5. Applicatios of the Basic Formula x x + x x + x +, x (5.3 x + x x, x < (5.4 + x x + x x + x, x, (5.5, x > (5.6 + x x x x ( + x x (x x, x, (5.7 ( x

22 Remar 5. The followig idetity is Problem 78 of Math. Magazie, Vol. 4, May 969, P. 53 x + x + x + x +, x > (5.8 x + x + Remar 5.3 I the followig two idetities, we assume r is a positive iteger. r r i ixir x r x r x r, x, r (5.9 x i0 xir r 3 x 3 + x 3 x, x < (5.0 + x 3 + x 3 x 6 Vadermode Covolutio ( x ta ( x cot cot x (5. 6. Itegral Versio of the Vadermode Covolutio Assume r ad q are oegative itegers. The, ( ( ( r q r + q. (6. i i i0 Equatio (6. is ow as the (itegral Vadermode Covolutio. 6.. Applicatios of Equatio (6. Remar 6. Throughout this sectio, we will assume r, q, ad are oegative itegers. Also recall that for real x, [x] is the iteger part of x. i0 ( ( r + q i r i ( r + q r (6.

23 ( ( ( ( ( ( ( 3 ( + ( ( (6.3 (6.4 (6.5 (6.6 Remar 6. The followig two idetities are from Problem 344, proposed by B. C. Wog, i The America Math. Mothly, March 930, Vol. 37, No. 3. ( ( ( (6.7 [ ] ( ( ( 6. Geeralized Versio of the Vadermode Covolutio, (6.8 Assume x ad y are arbitrary real or complex umbers. Assume is a oegative iteger. The, ( ( ( x y x + y. (6.9 Equatio (6.9 is the (geeralized Vadermode Covolutio. 6.. Specific Evaluatios of Equatio (6.9 Usig the Trasformatio ( ( (6.0 3

24 ( ( (6. ( ( ( ( ( ( ( ( { 0,, 0 { 0,, 0 ( ( ( ( ( ( + ( ( +, 0 4, 0, ( ( ( (6. (6.3 (6.4 (6.5 ( Various Applicatios of Equatio (6.9 Remar 6.3 For the rest of Chapter 6, we assume, uless otherwise specified, that α ad r are oegative itegers. We also assume x, y, ad z are arbitrary real or complex umbers. ( ( ( ( ( x y x y + x α α α α (6.7 Remar 6.4 I the followig two idetities, we assume f(x is a polyomial of degree, amely f(x i0 a ix i b x (. ( ( x y f( ( ( x x + y b (6.8 4

25 ( ( x y f( + z b ( ( ( z x y + x α α α α α0 (6.9 ( ( x + α α ( ( x (6.0 Remar 6.5 I the followig idetity, due to Laplace, assume u is a real or complex umber, u. ( x + u ( + u ( x + u (6. ( + u ( + (( x + x + x + ( x, x 0, (6. ( + (x + (x x +, x 0, (6.3 x ( ( ( x x ( (6.4 ( ( ( x ( ( + ( ( + ( + ( ( 4 + ( x + + (6.5 (6.6 ( ( x + ( ( + ( ( + ( ( x (6.7 ( ( + ( ( + ( ( + (6.8 5

26 ( ( x x ( ( x x ( (x ( x + + (6.9 ( ( x x ( ( x x + ( x, (6.30 Remar 6.6 The followig three idetities are foud i H. F. Sadham s Problem 459 [953, 47] of The America Math. Mothly, April 954, Vol. 4, pp ( ( x ( ( x ( + ( ( x ( x ( ( x ( x ( ( ( ( x + x + + r ( z y r ( x ( x (, (6.3, (6.3 ( x + (, (6.33 r ( z (y +, r (6.34 r r ( z x r r r ( z (x +, r (6.35 r ( x r+ r + ( x f( r ( r (x + ( r +, r (6.36 r + ( x ( ( f( + (6.37 6

27 ( x ( y ( + y ( x y (6.38 ( ( x y ( ( x ( y + (6.39 Remar 6.7 The followig idetity, due to H. L. Krall, is from Part of Problem 783, Math. Magazie, Vol. 44, 97, p.4. ( ( ( ( ( ( x y z x + y + z x + z y + z ( (6.40 ( ( ( ( ( y x + y + z x + z x + z y + z x x (6.4 ( x ( y z ( ( x+y+z ( x+z ( y+z ( x+y+z (6.4 ( x ( y x+y+z ( ( z + ( x+z ( y+z ( z ( Equatio (6.9 i Reciprocal Biomial Idetities ( ( z ( ( y 6.3. Specific Evaluatios of Equatio (6.44 ( y z ( y (6.44 ( ( z + ( ( x+ ( x z ( x (6.45 ( + ( ( x+ ( x+ ( x (6.46 7

28 ( ( z ( ( z ( ( ( x+ + + ( + ( x+z+ ( x+ (6.47 ( z + + ( (6.48 ( (6.49 Remar 6.8 The followig idetity is due to Harry Batema. The reader is referred to The Harry Batema Mauscripit Proect by Erdelyi, Magus, ad Oberhettiger. I particular, see Higher Trascedetal Fuctios, Vol. I, [Sectio.5.3, P. 86], McGraw-Hill, 953. ( ( z ( z ( ( z ( Applicatios of Equatio (6.44 ( z+ Remar 6.9 I the followig two idetities, we will assume x ad z are ot values which mae the deomiator equal to zero. ( ( z ( x+ ( ( + x ( ( x+ ( x+z+ ( x+ + x ( ( ( (x +, (6.5 + x +, (6.5 + x + z, (6.53 ( ( + ( (, (6.54 Remar 6.0 The followig idetity, due to R. R. Goldberg, is foud i Problem 4805 of The America Math. Mothly, Vol. 65, 958, P ( ( + ( + (, (6.55 8

29 6.4 Applicatio of Equatio (6.9 to ( ( ( x x ( ( ( ( x x ( ( ( x + x + ( ( ( ( ( ( x + x x ( ( Specific Evaluatios of Equatio (6.56 ( ( ( ( + 3 ( ( ( ( ( ( ( 3 + ( 5 ( ( ( ( ( 3 3 ( ( 4 ( 4 (6.57 (6.58 (6.59 ( ( ( ( 4 ( (6.60 ( ( ( ( + ( ( [ ] (6.6 ( ( ( + ( ( + ( ( 3 + (6.6 ( ( ( ( ( ( 4 ( + ( ( (6.63 9

30 6.4. Applicatios of Equatio (6.56 Remar 6. The followig idetity is foud i L. Carlitz s Note o a formula of Szily, Scripta Mathematica, Vol. 8 (95, pp r ( α ( ( r α r ( α ( r α ( α α r (6.64 ( r α α ( ( ( ( [ x x ( ( + x + ] ( [ ] [ ]( ( ( ( ( ( x x ( ( + x x (6.65 (6.66 ( ( ( x + x + ( ( ( + ( [ ] [ x + [ ]( ] (6.67 Remar 6. I the followig idetity, we assume x is a real or complex which is ot a egative iteger. We evaulate x! Γ(x +. ( ( ( x x ( ( + ( [ ] [ x + [ ]( ] ( x x ( x ( The Idex Shift Formula ad Equatio (6.9 ( ( ( x + x + α + α ( ( x α ( + x α (6.69 (

31 6.5. Applicatios of Equatio (6.69 Remar 6.3 For the followig two idetities, a referece is P. Tardy s, Sopra alcue formole relativa ai coefficieti biomiali, Battaglii s Joural, Vol. 3 (865, pp. -3. I the secod idetity, we assume a is a oegative iteger. ( ( ( x + x + x + ( ( ( x + x + a x + a ( x + + ( x + a + a ( x + + ( x + + a (6.7 (6.7 ( ( x z α0 ( ( x ( ( ( ( (z 4 α0 ( ( ( ( α ( x + α (z α (6.73 ( ( x + α ( α α (6.74 α α0 α0 ( α ( α α ( α ( α α ( α α ( α α (z α ( α (6.76 Remar 6.4 The followig idetity is from the Mathematical Reviews, Vol. 7, No 5., May 956, pp ( ( x + ( x (6.77 3

32 Remar 6.5 The followig idetity is from the Mathematical Reviews, Vol. 8, No., Jauary 957, P. 4. ( ( ( x + x + (6.78 ( ( ( x + x x + (6.79 ( ( x x ( x (6.80 Remar 6.6 For the followig idetity see Remar o a Note of P. Turá by T. S. Naudiah i America Math Mothly, Vol. 65, No. 5, May 958, P ( ( ( m x + y + x y x + m + ( ( x y, m itegeral, m 0 (6.8 m Remar 6.7 The followig idetity is a special case of Saalschütz s Theorem. See the review by L. Carlitz i Math Reviews, Vol. 8, No.. Jauary 957, P. 4. ( ( ( ( ( α x + + α x + α x + + α α ( ( ( x + r x r ( r α r ( ( ( x x ( ( r ( α + α r ( ( x ( x α (6.8 (6.83 (6.84 ( ( + ( ( 3 ( (6.85 3

33 ( ( 3 + ( ( ( 3 (6.86 ( ( 3 + ( ( ( ( 4 ( 3 (6.87 ( ( (6.88 ( ( ( + ( + ( (6.89 ( ( ( ( ( ( x x α + x α α α (6.90 ( ( ( ( x y ( +r ( x+y+ ( ( x+r+ ( y+ +r ( x+y+ ( 3 (6.9 (6.9 Remar 6.8 The followig idetity solves a problem o Page, Vol. 9, 947, of Nors Matematis Tidssrift. ( ( x ( y ( x+y+ ( x+ ( y+ ( x+y+ (6.93 ( ( ( ( ( x x α x + α + x + α + + α + α ( ( ( ( ( r x + + r + x + + r x + + r + r r (6.94 (

34 6.6 The Mius Oe Traformatio Applied to Equatio (6.9 ( ( ( x + y + x + y + + (6.96 β ( ( α β α 7 Alteratig Covolutios ( +, β α (6.97 α + β + Remar 7. Throughout this chapter, we assume, uless otherwise specified, that ad r are oegative itegers, while x, y, ad z are arbitrary real or complex umbers. 7. Evaluatio of ( ( x( x ( ( ( x x x ( ( Remar 7. I the followig idetity, we evaluate x! Γ(x +. (7. ( ( ( ( ( x + x x x ( ( + x x ( Specific Evaluatios ad Applicatios of Equatio (7. ( ( ( ( 4 (7.3 ( ( ( + ( ( (7.4 ( ( ( (7.5 34

35 ( ( ( ( ( ( (x x x ( + + ( x ( ( ( + ( + ( x+ 7. Evaluatio of ( ( x( x ( x+, (7.6 ( x Remar 7.3 Recall that for real x, [x] deotes the iteger part of x. ( ( ( x x ( ( + x ( [ ] [ ] ( ( x ( x ( ( + ( [ x ] (7.7 (7.8 (7.9 x [ ]( x ( x, see Remar 7. (7.0 ( ( x + ( x + ( x + ( + ( ( Applicatios of Equatio (7.9 ( ( y x + y ( r ( α0 α+[ α ] ( x α ( ( α + y [ α ] (7. r ( ( ( ( r + ( (, r (7.3 r r r ( ( [ x y ] ( ( x y x ( ( [ ] ( ( ( x x x ( ( (7.4 (7.5 35

36 Remar 7.4 I the followig two idetities, we evaluate z! by Γ(z+, wheever z is a real(complex umber which is ot a egative iteger. A referece for these two idetities is H. Batema s, Higher Trascedetal Fuctios, Vol I, Chapter II [editor W. Magus]. ( ( z ( z+ ( ( z z ( (z! π ( ( z! (7.6! ( z (z! π ( ( z! (7.7! [ ] ( ( x x ( x + ( [ ] ( x ( [ ] + (7.8 [ ] ( ( x + x ( x ( [ ] ( x ( [ ] +, (7.9 [ ] ( ( + ( [ ] ( ( [ ] + (7.0 [ ] ( + ( ( [ ] ( ( [ ] +, ( Quotiet Idetities Ivolvig Equatios (6.9 ad (7.9 Remar 7.5 Throughout this sectio, the reader may let x be a real or complex umber. The x! is evaluated as Γ(x First Example of a Reciprocal Idetity ( ( +x f( ( ( x ( +x +x x ( ( x x f( (7. 36

37 Specific Evaluatios of Equatio (7. ( ( +x +x x + + (x + ( x (7.3 x ( ( ( +x +x ( +[ ( ] x ( x+ + ( ( x x (7.4 ( ( ( x+ x+ ( x+ ( x ( x+ x ( ( x+ ( x+ x+ ( Secod Example of a Reciprocal Idetity ( ( ( ( + x ( + x ( +x +x ( +x ( ( ( +x ( +x (7.6 ( ( + x + x ( 4 + x + x ( Third Example of a Reciprocal Idetity ( ( + x f( + x Specific Evaluatios of Equatio (7.8 ( ( + x ( + x ( +x +x ( +x ( +x +x ( +x ( ( + x + x f( (7.8 ( [ ] + ( ( x (7.9 ( ( + x + x ( + x + x (

38 7.4 Evaluatio of ( ( y z ( ( ( y z ( ( 4 α ( ( y z y ( α α α α0 ( α α ( 4 4α α 8 A Itroductio to the th Differece Operator (7.3 α (7.3 Remar 8. Throughout this chapter, we will assume, uless otherwise specified, that ad r are oegative itegers, x is a real or complex umber, ad h is ozero real or complex umber. 8. Defiitio ad Basic Properties of h f(x 8.. Defiitio of h f(x Relatioship to Derivative h f(x d f(x dx lim h 0 ( + ( f(x + h h (8. ( f(x + h ( + (8. h h Liearity Properties of h h(f(x + ϕ(x hf(x + hϕ(x (8.3 Two Basic Examples h r hf(x r h hf(x +r h f(x (8.4 h(cf(x c hf(x, c is a costat (8.5 Remar 8. I the followig two examples, we assume a ad α are ozero real or complex umbers. ( a ha x a x h (8.6 h h( + αh x h ( + αh x h α (8.7 38

39 8.. Iversio of Equatio ( Calculatio of ( x f(x + h p! p x p ( ( hf(x (8.8 Remar 8.3 Throughout this sectio, we assume p is a iteger while x is a ozero real or complex umber. ( p ( x! p p x x ((x (+p p x p ( +! p (8.9 + x (+p ( ( x! x ( +! x ( x + Remar 8.4 The followig two idetities are foud i D. Steiberg s Combiatorial Derivatios of Two Idetities, Mathematics Magazie, Vol. 3, No.4, 958, pp m m!m (m! m, for m < + (8. m!m (m! + m! (m! m, for m > (8.! ( +! (8.3 (! ( + ( +! + + (8.4 ( x +! ( x ( x + ( +! x + ( (8.5 + x 39

40 ( +! ( + ( + ( +! + ( + (8.6 x + x(x! ( x x + ( +! ( x (8.7 + ( +! ( +! ( Euler s Traformatio If the ( f(0 ( f( 0 ( ( f(, (8.9 ( f(0. ( Alterative Defiitio of the th Differece Operator Defiitio of h Iversio of Equatio (8. h f(x + h ( ( f(x + h (8. ( ( hf(x (8. h f(x ( + ( f(x + h (8.3 + f(x + h ( + ( + hf(x (8.4 40

41 Examples of Equatios (8. ad (8. ( (, (8.5 ( (, (8.6 Example of Equatio (8.3 Let f( ( ( F (. (8.7 The, f(i i ( + ( F (. (8.8 + Geeralizatio of Equatios (8. ad (8.: The Iversio Pair Theorem ( ( f( ( g(α (8.9 α α if ad oly if ( ( g( ( f(α (8.30 α α 4

42 8. The Shift Operator E h 8.. Defiitio of E h E h f(x f(x + h ( Relatioships Betwee the Shift ad Differece Operators ( ( hf(x ( E h hf(x (8.3 hf(x h E h f(x ( ( E h f(x (8.33 ( h hf(x (8.34 E h hf(x he hf(x 8..3 Product Formula for h uv f(x + h f(x + h h (8.35 huv ( hu h (Ehv (8.36 Applicatio of Equatio (8.36 Remar 8.5 I the followig example, we assume a is a ozero real or complex umber. h(e ax f(x e ax ( ( e e ah ah hf(x (8.37 h 8..4 Recurrece Formula for hf(x + h hf(x + h h + h f(x + hf(x (8.38 4

43 9 Calculatios Ivolvig ( x xr Remar 9. Throughout this chapter, we assume ad m are oegative itegers. We also assume, uless otherwise specified r, z, x, ad a are real or complex umbers for which the Gamma fuctio is defied. (!(r! ( β(r, + + r ( + r! (9. ( ( a + ( + (9. 9. Specfic Evaluatios of Equatio (9. ( ( + a ( + ( (9.3 ( ( +! ( ( +! (9.4 ( (! ( +! + ( Applicatios of the Iversio Pair to Equatio (9. ( ( ( + a a +, a 0 (9.6 ( ( ( +m m + m ( ( ( (9.7 (9.8 43

44 ( ( + r + r ( (9.9 a + + a 9.. Applicatios of Equatio (9.7 ( ( + + ( + a, a 0 (9.0 ( ( ( (9. +a a ( ( + a ( +a (9. ( ( ( m ( ( + (! z + e z a + a a, a 0 (9.3 ( +m ( z+,, m (9.4, z 0 (9.5! ( (! + +! e ( + (! e!!! (9.6 (9.7 x + ( z+ ( z z + ( + x, x <, x x x < (9.8 44

45 Two Restatemets of Equatio (9.8 x + z + x + ( z ( x + x Three Specific Evaluatios of Equatio (9.8 ( z ( z+ x, x < (9.9 ( + x z + x + z +, x + x < (9.0 ( z+ z + (z + (9. ( x+ ( x x + (9. ( x + x ( x+ 0 Euler s Fiite Differece Theorem, x 0 (9.3 + Remar 0. Throughout this chapter, we assume,, m, p, α, ad r are oegative itegers. We also assume x, z, a, y, ad b are real or complex umbers. 0. Statemet of Euler s Fiite Differece Theorem ( ( { 0, 0 < (!, (0. 0. Variatios of Equatio (0. ( ( + ( ( +! ( ( + ( (3 + ( +! 4 (0. (0.3 45

46 0.3 Polyomial Extesio of Equatio (0. Remar 0. Give a polyomial r a of degree r i, with a free of, Equatio (0. implies r ( { a ( 0, r < (0.4 (!a, r 0.3. Applicatios of Equatio (0.4 ( { ( (x r 0, r <!, r (0.5 ( { ( (a b r 0, r < b!, r (0.6 Remar 0.3 Througout the remaider of this chapter, we will assume, uless otherwise specified, that f(x is a polyomial of degree i x, amely f(x i0 a ix i. m ( m ( { 0, < m f(x z a m z m m!, m (0.7 ( ( { m ( 0, r < r ( m, r, m ot ecessarily a iteger (0.8 ( ( { ( 0, r < r (, r (0.9 Two Applicatios of Equatio (0.9 p p ( + ( + ( ( p, p, (0.0 p p ( ( ( ( ( + p, (

47 ( ( r + ( r ( ( y + x ( r { 0, r < (, { 0, r < r ( x, r, (0. (0.3 ( ( ( x α ( x α { 0, 0 α < ( x (, α (0.4 ( ( x + ( r ( x ( r (0.5 ( ( { α + ( 0, α < α ( ( α (0.6, α ( ( x x r r (0.7 ( ( ( x + x ( ( r + r + ( ( ( x + α ( x + r α r Three Applicatios of Equatio (0.5 ( ( x ( ( r ( α + α r ( x ( x α (0.8 (0.9 (0.0 47

48 Remar 0.4 The followig two idetities are due to the wor of H. Batema ad T. Oo. For refereces see Questio i Nouv. A. de Math, Vol.4. No. 4, 95, pp. 9-9, ad Problem 38, Nouv. A. de Math, Vol.4. No. 5, 95, pp Also see Notes o Biomial Coefficiets by H. Batema, P. 54. a ( a ( a + a+ ( a+ ( a a (, a a oegative iteger, a (0. a ( a a ( ( ( a a a (, a a oegative iteger, a (0. ( (y (x y x, (0.3 ( (y (x y x, (0.4 ( (x x, (0.5 Applicatio of Equatio (0.4 a x f(x ( + (x + y + y ( y y a (0.6 + Abel s Sum (x + y + z x ( (y + ( z (x + z (0.7 Applicatio of Equatio (0.4 to a Expasio of e xz e xz (x + y z e yz x! (0.8 48

49 Remar 0.5 The followig idetity is M. S. Klami s solutio to Problem 4489 [95, 33] of The America Math. Mothly, Sept. 953, P ((x + yw! exz wx, w < y e, x 0 (0.9 (yw! z w, w ze yz (0.30 (xe x! e x (0.3 ( e! e (0.3 Extesio of Equatio (0.3! (xe x ex x, x (0.33 x ( x+! e! x + (0.34 Applicatio of Equatio (0.33 ( ( ( +!! (

50 Three Applicatios of Equatio (0.35 Remar 0.6 The followig idetity is from Problem E38 of The America Math, Mothly, Vol. 65, 958, P Recall that for real x, [x] deotes the floor of x.! [e!], (0.36! ( ( [e!] (! + (0.37 ( ( ( + ( [e!], (0.38 ( + e! [e!], (0.39! [e!] [e( +!], (0.40 Newto-Gregory Expasios for Polyomials Remar. Throughout this chapter, we will assume f(x is a polyomial of degree, amely, f(x i0 a ix i. We will also assume, uless otherwise specified, that x ad y are real or complex umbers, while is a oegative iteger. Newto-Gregory Expasios f(x + y f(x + y ( x f(z zy (. ( x h hf(y, where h is a ozero real or complex umber (. 50

51 Recursive Formula for Fuctios Defied by th Differeces Remar. Throughout this chapter, we will assume is a oegative iteger. We also let, for real x, [x] deote the floor of x.. First Recursive Formula If the [ ] ( f( F ( (, (. [ ] ( ( f( + + f(0 F ( +. (. + Note that f( idepedet of.. Secod Recursive Formula If F ( ( ( f(, (.3 the ( f( + ( + f(0 F ( +. (.4 + Note that f( idepedet of. 5

52 .3 Third Recursive Formula Remar. Throughout this sectio, we assume r ad α are itegers. Defie The, Also, F (r + ( f( + r + (, r. (.5 ( +r+ ( F (r + r + + ( + f(r ( + f( + r. (.6 ( +r ( F (α + ( α ( α + α ( + α f( + +α F +α ( ( + α ( α α + +α ( + α f( + (, α. (.7 + ( +α α.3. Applicati0s of Equatio (.7 α F ( + ( ( + ( ( f( + + (.8 Remar.3 The followig idetity, due to H. F. Sadham, is from Advaced Problem 459 of The America Math. Mothly, Jauary 953, Vol. 60, No., P. 47. ( ( ( ( + ( ( +! + ( ( + ( +! + ( + ( ( + ( ( ( + α + α + ( + 4+ ( ( + ( (.0 + (, 0, 0, 4 ( + ( (. α is a real or complex umber (. 5