Legendre Polynomials

Index

Symbols upper r φs q, x (q-hypergeometric se- P (x) (Legendre polynomials), 1 lower n ries), 26 := (definition), 1 (a , a ,...,a ; q) (q-Pochhammer nota- Γ(z) (Gamma function), 1 1 2 r k tion), 27 C (field of complex numbers), 1 ( ; ) a q k (q-Pochhammer symbol), 27 Rez (Real part of z), 1 [k]q (q-brackets), 28 t=b [k]q ! (q-factorial), 28 F(t) = F(b) − F(a) (difference), 1 t=a n ! (q-binomial coefficient), 28 k (factorial), 1 k q N 0 (set of nonnegative integers), 1 Γq (z) (q-Gamma function), 28 Z (ring of integers), 2 θ f (x) = xf(x) (differential operator), 29 R (field of real numbers), 2 x (floor function), 31 N (set of positive integers), 2 Ln(x) (Laguerre polynomials), 57 Δ (z)k (shifted factorial, Pochhammer sym- (forward difference operator), 64, 141 bol), 3 pn(x|q) (little q-Legendre polynomials), 71 P (x; c; q) (big q-Legendre polynomials), Res f (z) (residue of f at z0), 3 n z=z0 71 z ( | ) (z choose k, binomial coefficient), 5 Pn x q (continuous q-Legendre polynomi- k als), 71 ( ,w) B z (Beta function), 5 Pn(x; q) (continuous q-Legendre polynomi- K (field of characteristic zero), 12 als), 71 (α) Q = Q(x1, x2,...,xm ) (extension of field Ln (x) (generalized Laguerre polynomi- of rational numbers), 12 als), 71 (α) upper Ln (x; q) (q-Laguerre polynomials), 72 p Fq x (generalized hypergeo- lower Fn (Fibonacci numbers), 74 metric function), 12 fn(x) (Fasenmyer polynomials), 75 , a b Zn(x) (Bateman polynomials), 75 2 F1 x (Gauss hypergeometric c Hn(x) (Hermite polynomials), 76 function), 14 gcd( f, g) (greatest common divisor of poly- a nomials), 80 1 F1 x (Kummer’s confluent hyper- b deg (qk ) (degree of polynomial), 82, 87 geometric function), 14 disp (qk , rk ) (dispersion of polynomials), 82 K(k) (field of rational functions over K), 14 max S (maximum of set S), 82 K[ ] K k (ring of polynomials over ), 14 Hk (harmonic numbers), 89 m ak |k=n (substitution), 20 k (falling factorial), 95

W. Koepf, Hypergeometric Summation, Universitext, 271 DOI: 10.1007/978-1-4471-6464-7, © Springer-Verlag London 2014 272 Index k!! (double factorial), 97 q-differential equations, 219 ζ(z) (zeta function), 126 q-Gosper, 94 K, N (shift operators), 135 q-Petkovšek, 196 Kn(x; p, N) (Krawtchouk polynomials), q-Zeilberger, 138, 139 141 Risch, 79, 236 Mn(x; β,c) (Meixner polynomials), 141 Risch-Bronstein, 227 Cn(x, a) (Charlier polynomials), 141 simpcomb, 16 Qn(x; α, β, N) (Hahn polynomials), 141 Zeilberger, 117, 127 ∇ (backward difference operator), 141 Almkvist-Zeilberger algorithm, 227, 239 2 Wn(x ; a, b, c, d) (Wilson polynomials), Andrews’ identity, 157 141 Antidifference, 79 (α,β)( ) Pn x (Jacobi polynomials), 146 m-fold, 153 ( ) Jn x (Bessel functions), 147 Apéry numbers, 31, 125, 145, 185 Dn (number of derangements), 199 Apéry recurrence equation, 126, 175, 185 D (unit disk), 212 Apparent singularity, 194 D q (q-derivative operator), 218, 262 a priori bound, 60, 87, 134 ( ), ( ) eq x Eq x (q-exponential functions), 219 Askey-Gasper identity, 132 ( ), ( ) sinq x Sinq x (q-sine functions), 219 Askey-Gasper inequality, 132 ( ), ( ) cosq x Cosq x (q-cosine functions), 219 Askey-Wilson scheme, 72, 138 ν ( ) Cn x (Gegenbauer polynomials), 221 Associated Legendre functions, 222 ( ) Fn t (Bateman functions), 221 assume, 8 m ( ) Pn x (associated Legendre functions), 222 asympt, 193 (α)( ) Bn x (Bessel polynomials), 223 Asymptotic series, 193 erf (x) (error function), 232 Axiom, viii D (differential operator), 245 A(n, x) (Abramowitz functions), 249 Ai(x) (Airy function), 251 Bi(x) (Airy function), 251 B Backward antidifference, 81 CTz F(z) (constant term of Laurent polynomial), 267 Backward difference operator, 141, 258 Bailey identity, 107, 161 Pn(x; a, b, c; q) (big q-Jacobi polynomials), 268 Bailey transformation, 148 Bailey’s hypergeometric database, 36 Balanced hypergeometric series, 47 A Basic hypergeometric function, 27 Abramowitz functions, 249 Bateman functions, 221, 252, 264 add (addition), 55, 112 Bateman integral representation, 248 Adjoint operator, 182 Bateman polynomials, 75 Admissible hypergeometric term, 167 Bessel differential equation, 212, 222 Airy functions, 251 Bessel functions, 147, 212, 222, 251 Airy integral, 251 Bessel polynomials, 223 Algebra of q-holonomic functions, 220 Beta function, 5, 244 Algebra of holonomic functions, 189 Bieberbach conjecture, 31, 132, 221 algebraicrechyper(rec,s(n)), Weinstein proofs, 132 187 Big q-Legendre polynomials, 71, 148 Algorithm Big q-Jacobi polynomials, 268 Almkvist-Zeilberger, 227, 239 Bilateral sum, 13 continuous Gosper, 227 bind(FormalPowerSeries), 222 extended_gosper, 154 binomial(n,k), 7 Fasenmyer, 54 Binomial sum identity, 12 Gosper, 79 Binomial theorem, 28, 90, 111 van Hoeij, 169, 190 q-analogue, 28, 115, 148 Petkovšek, 169 Branges’ theorem, de, 31, 132, 221 Index 273

C D CAOP project (www.caop.org), 138 Database catch, 111 of antiderivatives, 79 Cauchy integral formula, 255 of hypergeometric identities, 36 Certificate recurrence equation, 61, 65 of q-hypergeometric identities, 44 Certificate, rational Decision procedure, 60 of Almkvist-Zeilberger algorithm, 240 Definite integration, 227, 239 Definite summation, 12, 103, 117 of continuous Gosper algorithm, 237 natural bounds, 13, 24 of extended Gosper algorithm, 167 non-natural bounds, 24, 128 of extended WZ method, 159 degreebound(p,q,r,k), 93, 96 of hypergeometric term, 91 deltarodriguesdiffeq(g,h,n,s(x)), of q-WZ method, 113 267 of WZ method, 107 deltarodriguesrec(g,h,x,s(n)), of Zeilberger algorithm, 127 267 Charlier polynomials, 141, 259 Derangement numbers, 199 generating function, 267 Derivative rule, 215, 250 Rodrigues formula, 260 Bateman functions, 221 checksum(F,R,k,n), 114 Bessel functions, 222 Chu-Vandermonde identity, 36, 107 Bessel polynomials, 223 Classical discrete orthogonal polynomials, Hermite polynomials, 222 140, 266, 267 hypergeometric function, 29 Classical orthogonal polynomials, 223 Laguerre polynomials, 222 Clausen formula, 130 Legendre polynomials, 216, 221 de2diffop, 246 Clausen identity, 114 DEtools package, 246 Clausen product identity, 130, 144 DEtools[de2diffop], 246 Closedform(F,k,n), 130 DEtools[DFactor], 246 closedform(F,k,n), 122 DEtools[diffop2de], 247 Companion identity, 112 DEtools[mult], 247 Compatible recurrence equations, 203 DFactor, 246 Complexity, 60, 144, 188, 199, 201 diff(f,x), 133 Confluence process, 27, 32 diffeqtorec, 201 Constant term of Laurent polynomial, 267 Difference operator, 64, 95, 141, 258 contdegreebound(p,q,r,x), 232 Differential equation contdispersionset(q,r,x), 232 Abramowitz functions, 249 contfindf(p,q,r,x), 233 Airy functions, 251 contgosper(f,x), 234 Bateman functions, 221 Continuous q-Legendre polynomials, 71, Bessel functions, 212, 222 148 Bessel polynomials, 223 Continuous Gosper algorithm, 227 generalized Laguerre polynomials, 257 Contour integration, 255 Hermite polynomials, 223 holonomic, 205 contratio(f,x), 232 hypergeometric, 29 contupdate(p,q,r,x), 232 inhomogeneous, 212, 241 convert , 7 Jacobi polynomials, 223 ... convert( ,FormalPowerSeries), Laguerre polynomials, 223 218, 222, 262 Legendre polynomials, 209, 257 Cosine function, q-analogue, 219 Differential operator, 245 Creative symmetrizing, 202 diffop2de, 247 Critical point of hyperexponential term, 206 Discrete orthogonal polynomials, 140, 266, CTdiffeq(F,z,s(x)), 268 267 CTrecursion(F,z,s(n)), 268 Discrete Rodrigues formula, 258 274 Index

Dispersion of polynomials, 82 Fixed point free permutations, 199 dispersionset(q,r,k), 85, 93 Floor function, 31 Dixon identity, 20, 36, 64, 107 for loop in Maple, 42 q-analogue, 115, 148 FormalPowerSeries package, 218, Double sum identity, 131 222, 262 Dougall identity, 107, 122 Forward antidifference, 79 q-analogue, 115, 148 Forward difference operator, 64, 95, 141, 258 dsolve, 246, 247, 252 Fuchs relations, 192, 195 Dual identity, 112 Dummy variables in Maple, 42 Duplication formula of Γ function, 18 G GAMMA(z), 7 E Gamma function, 1, 241 _Envdiffopdomain, 246 duplication formula, 18 Equivalent recurrence equations, 203 reflection formula, 4 error , 111 Gauss hypergeometric function, 14, 32 Error function, 232, 237 Bateman integral representation, 248 Euler integral representation, 247 Euler integral representation, 247 Euler transformation, 213 Gauss identity, 36, 107, 161 Euler-Mascheroni constant, 5 Gcd, 80 expand, 7, 14, 32, 134 Exponential function, q-analogue, 219 Gegenbauer polynomials, 221 Exponential generating function, 261 generating function, 265 Charlier polynomials, 267 Generalized hypergeometric function, 12 Hermite polynomials, 265 Generalized Laguerre polynomials, 71, 74, Legendre polynomials, 268 222, 266 extended+_+gosper algorithm, 184 generating function, 262 extended_gosper(a,k), 156 parameter derivative, 145 extended_gosper(a,k,m), 156, 167 Rodrigues formula, 257 extended+_+sumrecursion, 168 generateproducts(f), 183 extended WZ certificate, 159 Generating function, 260 Charlier polynomials, 267 exponential, 261 F Gegenbauer polynomials, 265 factor, 83 generalized Laguerre polynomials, 262 factorial(k), 7 Krawtchouk polynomials, 267 Factorial part, 59, 81, 134 Legendre polynomials, 266 Factorization Meixner polynomials, 267 noncommutative, 173 Generation of identities, 123 rational, 22, 83, 187 Gessel-Stanton identities, 158 factor+_+over+_+Q, 184 GFdiffeq(F,a,z,n,s(x)), 261 Falling factorial, 95 GFrecursion(F,a,z,s(n)), 261 fasenmyer(f,k,s(n),nmax), 57 gfun Fasenmyer algorithm, 54 package, ix, 189, 201, 218 fasenmyerdiffeq(f,k,s(x),xmax), gfun[diffeqtorec], 201 73 gfun[‘rec*rec‘], 182 Fasenmyer polynomials, 75 gfun[‘rec+rec‘], 189 Favard’s Theorem, 75 gfun[rectodiffeq], 218 Fibonacci numbers, 74, 144, 266 Gosper findf(p,q,r,k), 96 -summable, 80 find_mfold, 167 algorithm, 79 Finite support, 12, 103, 117, 157 continuous version, 227 FirstWeyl (Singular), 245 q-analogue, 94 Index 275 gosper(a,k), 96, 111 term, 12 Greatest common divisor, 80 admissible, 167 Greatest factorial factorization, 93 antidifference, 80 Gröbner basis, 175 local type, 191 local type at ∞, 192 m-fold, 153 H (m, l)-fold, 157 Hahn polynomials, 141 proper, 59, 134 Rodrigues formula, 266 rational certificate, 91 Harmonic numbers, 89, 202 with respect to two variables, 103 Hermite polynomials, 76 transformation, 43, 213 derivative rule, 222 q-analogue, 148 differential equation, 223 hypergeomsols, 190, 193, 200 exponential generating function, 265 hyperrecursion(upper,lower,x,s(n)), Rodrigues formula, 265 166 Holonomic hyperterm(upper,lower,x,k), 26, differential equation, 205 31 function, 205 operator, 135 q-function, 220 I recurrence equation, 52 Identity HolonomicDE, 222 Andrews, 157 HolonomicRE(term,s(k)), 188 Askey-Gasper, 132 hsum package, ix, 15, 42, 43, 93, 130, 156, Bailey, 107, 161 166 binomial sum, 12 Hyperexponential term, 73, 205, 227 Chu-Vandermonde, 36, 107 antiderivative, 227 Clausen, 114 strictly, 206 Dixon, 20, 36, 64, 107 hypergeom(upper,lower,x), 41 double sum, 131 Hypergeometric Dougall, 107, 122 database Gauss, 36, 107, 161, 251 Bailey, 36 generation of, 123, 127 differential equation, 29 Gessel-Stanton, 158 function hypergeometric, 12 basic, 27 integral sum, 249 confluent, 14, 32, 250 Jackson, 115, 148 derivative rule, 29 Kummer, 36, 107, 223 Gauss, 14, 32, 247, 248 Pfaff-Saalschütz, 36, 107 generalized, 12 proving, 106, 123, 214, 249 Kummer, 14, 32, 250 q-Chu-Vandermonde, 48, 76, 113, 115, recurrence equation, 29 139, 148, 196 identity, 12 q-Dixon, 115, 148 series, 12 q-Dougall, 115, 148 balanced, 47 q-Gauss, 44, 48 basic, 27 q-Kummer, 115, 148 generalized, 12 q-Pfaff-Saalschütz, 44, 76, 115, 139, 148 k-balanced, 47 Stanley, 45 lower parameters, 13 Strehl, 70 nearly-poised, 47 Székely, 40 Saalschützian, 47 Watson, 36, 107, 161 upper parameters, 13 Whipple, 36, 107, 161 well-poised, 47 Indefinite integration, 227 sum, 12 Indefinite summation, 80 276 Index

Inequality of Askey and Gasper, 132 KoepfZeilberger, 167 infhsum package, 138 Krawtchouk polynomials, 141 infinitetype, 193 generating function, 267 infolevel, viii, 92 Rodrigues formula, 266 Inhomogeneous Kummer hypergeometric function, 14, 32 differential equation, 212, 241 integral representation, 250 recurrence equation, 129, 241 Kummer identity, 36, 107 int(f,x), 133, 235 q-analogue, 115, 148 int(f,x=a..b), 7 Kummer transformation, 46, 213 intdiffeq(F,t,S(x)), 242 intdiffrule(F,t,S(n,x)), 250 Integer-linear, 12 L Integral formula of Cauchy, 255 Laguerre polynomials, 57 Integral representation differential equation, 223 Bateman, 248 generalized, 71, 74, 222 Bateman functions, 252 derivative rule, 222 Bessel functions, 251 generating function, 262 Beta function, 5, 244 parameter derivative, 145 Euler, 247 Rodrigues formula, 257 Gamma function, 1, 241 q-analogue, 72, 148, 262 generating function, 261 Legendre functions, associated, 222 Kummer hypergeometric function, 250 Legendre polynomials, 23, 46, 125, 220 Legendre polynomials, 256 derivative rule, 216, 221 Rodrigues formula, 255 differential equation, 209 Integral sum identity, 249 exponential generating function, 268 Integration generating function, 266 contour, 255 integration rule, 218 definite, 227, 239 q-analogues, 76 indefinite, 227 recurrence equation, 52, 73 Integration rule, 216 Rodrigues formula, 256 Bessel polynomials, 223 Schläfli’s integral, 251, 256 Jacobi polynomials, 218, 223 Little q-Legendre polynomials, 71, 148 Legendre polynomials, 218 recurrence equation, 76 intrecursion(F,t,S(n)), 242 LocalInfiniteType(RE,s(n)), 202 isolve, 83 Local type of hypergeometric term, 191 IsZApplicable, 138 at ∞, 192 Lower parameters of hypergeometric series, 13 J LREtools[hypergeomsols], 190, Jackson identity, 115, 148 193, 200 Jacobi polynomials, 146, 223 integration rule, 218, 223 parameter derivative, 147 M q-analogue, 268 Macsyma, viii Rodrigues formula, 265 map, 55 Mathematica, ix MAXORDER, 124 K Meixner polynomials, 141 k-balanced hypergeometric series, 47 generating function, 267 kfreediffeq(f,k,x,kmax,xmax), Rodrigues formula, 266 73 m-fold antidifference, 153 kfreerec(f,k,n,kmax,nmax), 55 m-fold hypergeometric term, 153 k-free recurrence equation, 51 Monic polynomial, 170 KoepfGosper, 157 (m, l)-fold hypergeometric term, 157 Index 277

Mgfun package, ix big q-Legendre, 71, 148 mul (multiplication), 184, 197 big q-Jacobi, 268 mult, 247 Charlier, 141, 259 multsum package, 64 continuous q-Legendre, 71, 148 Fasenmyer, 75 Gegenbauer, 221, 265 N Hahn, 141 nablarodriguesdiffeq(g,h,n,s(x)), Hermite, 76, 222, 223, 265 260 Jacobi, 146, 218, 223, 265 nablarodriguesrec(g,h,x,s(n)), Krawtchouk, 141 259 Laguerre, 57, 71, 74, 145, 222, 223, 257, Natural bounds of definite sum, 13, 24 262, 266 ncfactor library (Singular), 245 Legendre, 23, 46, 52, 73, 125, 209, 216, ncpoly package (REDUCE), ix, 175, 245 218, 220, 221, 251, 256 Nearly-poised hypergeometric series, 47 little q-Legendre, 71, 76, 148 Newton polygon, 195 Meixner, 141 NewtonpolygonRE(RE,s(n)), 195 monic, 170 Noncommutative factorization, 173, 181 orthogonal, 223 Non-natural bounds, 128 discrete, 140, 266, 267 normal, 15, 32, 55, 133, 236 q-Laguerre, 72, 148, 262 Number of derangements, 199 Wilson, 141 Power series, 13 primedispersion(q,r,k), 84 O Probability integral, 8 Operator Proper hypergeometric term, 59, 134 differential, 245 Proving identities, 106, 123, 157, 214, 249 holonomic, 135 shift, 135 Optional argument of a Maple procedure, 56 Ordinary differential equation, 205 Q Orthogonal polynomials q-analogue, 28 classical, 223 q-binomial coefficient, 28 classical discrete, 140, 266, 267 q-binomial theorem, 28, 115, 148 q-brackets, 28 q-Chu-Vandermonde identity, 48, 76, 113, P 115, 139, 148, 196 Parameter derivative q-cosine function, 219 Jacobi polynomials, 147 q-derivative operator, 218 Laguerre polynomials, 145 q-Dixon identity, 115, 148 Pascal triangle, 49, 56 q-Dougall identity, 115, 148 Petkovšek algorithm, 169 q-exponential function, 219 q-analogue, 196 q-factorial, 28 Pfaff transformation, 43, 213 q-Gamma function, 28 Pfaff-Saalschütz identity, 36, 107 q-Gauss identity, 44, 48 pochhammer(a,k), 123 q-Gosper algorithm, 94 pochhammer, 7 q-hypergeometric Pochhammer symbol, 3 database, 44 Polynomial part, 59, 81, 134 function, 26 Polynomial resultant, 83, 228, 234 series, 26 Polynomial solutions of recurrence equa- term, 27 tions, 169 transformation, 148 Polynomials q-Kummer identity, 115, 148 Bateman, 75 q-Laguerre polynomials, 72, 148 Bessel, 223 Rodrigues formula, 262 278 Index q-Legendre polynomials Rational factorization, 22, 83, 187 big, 71, 148 Rational-linear arguments, 12, 163 continuous, 71, 148 read, 15 little, 71, 76, 148 ‘rec*rec‘, 182 q-numbers, 28 ‘rec+rec‘, 189 q-Petkovšek algorithm, 196 rechyper(rec,s(n)), 187, 201 q-Pfaff-Saalschütz identity, 44, 76, 115, 139, recpoly(rec,s(n)), 200 148 rectodiffeq, 218 q-Pochhammer symbol, 27 rec2hyper(rec,s(n)), 183 q-sine function, 219 rec2poly(rec,s(n)), 175 q-Vandermonde identity, 48, 76, 113, 115, Recurrence equation 139, 148, 196 Apéry numbers, 126, 175, 185 q-WZ method, 113 discrete orthogonal polynomials, 140, q-Zeilberger algorithm, 138, 139 266 qbinomial(n,k,q), 44 holonomic, 52 qbrackets(k,q), 44 hypergeometric function, 29 qfactorial(k,q), 44 inhomogeneous, 129, 241 qfasenmyer(term,q,k,s(n),km,nm), k-free, 51 72 Laguerre polynomials, 58 qFPS package, 197, 198 Legendre polynomials, 52, 73 qGAMMA(k,q), 44 little q-Legendre polynomials, 76 qgosper(term,q,k), 94 ‘recursion/compare‘, 203 qHypergeomSolveRE, 198 REDUCE, 175, 245 qMultiplyRE, 197 Reflection formula of Γ function, 4 qphihyperterm(upper,lower,q,x,k), resultant(q,r,k), 83 44 Resultant of polynomials, 83, 228, 234 qpochhammer(a,q,k), 44 Risch algorithm, 79, 236 qratio(expr,k), 44 Risch-Bronstein algorithm, 227 qrecsolve(rec,q,s(n)), 196 Rodrigues formula, 255 qREtoqRE, 197 big q-Jacobi polynomials, 268 qrodriguesdiffeq(g,h,q,n,s(x)), Charlier polynomials, 260 268 discrete, 258 qshift, 197 generalized Laguerre polynomials, 257 qsimpcomb(expr), 44 Hahn polynomials, 266 qsum package, 44, 72, 94, 113, 139, 196, Hermite polynomials, 265 219 Jacobi polynomials, 265 qsumdiffeq(F,q,k,S(x)), 219, 268 Krawtchouk polynomials, 266 qsumrecursion(F,q,k,S(n)), 139 Legendre polynomials, 256 Quadratic transformation, 223 Meixner polynomials, 266 qWZcertificate(F,k,n), 113 q-Laguerre polynomials, 262 rodriguesdiffeq(g,h,n,s(x)), 256 R rodriguesrecursion(g,h,x,s(n)), ratio(a,k), 26 256 Rational certificate of Almkvist-Zeilberger algorithm, 240 of continuous Gosper algorithm, 237 S of extended Gosper algorithm, 167 Saalschütz identity, 36, 107 of extended WZ method, 159 Saalschützian hypergeometric series, 47 of hypergeometric term, 91 Schläfli’s integral, 251, 256 of q-WZ method, 113 seq(f,j=list), 42 of WZ method, 107 Shift operator, 135 of Zeilberger algorithm, 127 Shift structure, 82 Index 279

Shifted factorial, 3 T simpcomb(expr), 26 Telescoping sum, 61, 64, 104, 106, 119, 128 simpcomb algorithm, 16 termtohyper(f,k), 43 simplify, 7, 15 time(), 85 simplify_gamma algorithm, 16 Transformation simplify_power algorithm, 16 Bailey, 148 Sine function, q-analogue, 219 Euler, 213 Singular, 245 hypergeometric, 213 Singularity q-analogue, 148 apparent, 194 Kummer, 213 structure of hypergemetric term, 191, Pfaff, 43, 213 192 quadratic, 223 solve, 52 Watson, 148 Stanley identity, 45 Whipple, 142, 223 Stanton conjecture, 148 try, 111 Stirling formula, 97 Strehl identity, 70 Strictly hyperexponential term, 206 U Structure set, 62 update(p,q,r,k), 96 sum(a,k), 92 Upper parameters of hypergeometric series, sum(a,k=m..n), 25, 41 13 sumdeltanabla(F,k,s(x)), 141 ‘sumdelta+nabla‘(F,k,s(x)), V 141, 142 Vandermonde identity, 36, 107 sumdiffeq(F,k,S(x)), 210 van Hoeij algorithm, 169, 190, 197, 200, 218 sumdiffrule(F,k,S(n,x)), 215 Verbaeten completion, 62, 65, 67 sumintrule(F,k,S(n,x)), 217 Summable, Gosper-, 80 Summation W by parts, 96 Watson identity, 36, 107, 161 definite, 12, 103, 117 Watson transformation, 148 natural bounds, 13, 24 Weierstrass product representation, 4 non-natural bounds, 24, 128 Weinstein proof of Bieberbach conjecture, indefinite, 80 132 sumrecursion(F,k,s(n)), 124, 145, Well-poised hypergeometric series, 47 166 Whipple identity, 36, 107, 161 sumrecursion(F,k=a..b,s(n)), Whipple transformation, 142, 223 130 Wilf-Zeilberger method, 106 Sumtohyper(f,k), 42 Wilson polynomials, 141 SumTools package, 92, 138, 157, 167 www.caop.org, 138 SumTools[Hypergeometric] WZ certificate, 107 [Gosper], 92 extended, 159 SumTools[Hypergeometric] q-analogue, 113 [IsZApplicable], 138 WZcertificate(F,k,n), 111 SumTools[Hypergeometric] WZcertificate(F,k,n,m,l), 160 [KoepfGosper], 157 WZ method, 106 SumTools[Hypergeometric] q-analogue, 113 [KoepfZeilberger], 167 sum2qhyper(expr,q,k), 44 Support, finite, 12, 103 Z Symmetric product, 182, 196 zeilberger(F,k,s(n)), 121 Symmetrizing, 202 Zeilberger algorithm, 117, 127 Székely identity, 40 q-analogue, 138, 139