# Calculus of Finite Differences

American Mathematical Soc., 1965 - 654 pages
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### Table des matières

 On Operations 1 Historical and Bibliographical Notes 1 Definition of differences 3 Operation of displacement 5 Operation of the mean 6 Symbolical Calculus 7 Symbolical methods 8 Receding Differences 14 Central Differences 15
 The MaclaurinEuler summation formula 260 Bernoulli polynomials of the second kind 265 Symmetry of the Bernoulli polynomials of the second kind 268 Extrema of the polynomials 269 Particular cases of the polynomials 272 Operations on the Bernoulli polynomials of the second kind 275 Expansion of the Bernoulli polynomials of the second kind 276 Application of the polynomials 277

 Divided Differences 18 Generating functions 20 General rules to determine generating functions 25 Expansion of functions into power series 29 Expansion of functions by aid of decomposition into partial fractions 34 Expansion of functions by aid of complex integrals 40 Expansion of a function by aid of difference equations 41 Functions important in the Calculus of Finite Differences 16 The Factorial 45 The Gamma function 53 Incomplete Gamma function 56 The Digamma function 58 The Trigamma function 60 Expansion of log I x+1 into a power series 61 The Binomial coefficient 62 Expansion of a function into a series of binomial coefficients 74 Beta functions 80 Incomplete Beta functions 83 Exponential functions 87 Trigonometric functions 88 Alternate functions 92 Functions whose differences or means are equal to zero 94 Product of two functions Means 98 Inverse Operation of Differences and Means Sums 32 Indefinite sums 100 Indefinite sum obtained by inversion 103 Indefinite sum obtained by summation by parts 105 Summation by parts of alternate functions 108 Indefinite sums determined by difference equations 109 Differences sums and means of infinite series 110 Inverse operation of the mean 111 Other methods of obtaining inverse means 113 Sums 116 Sums determined by indefinite sums 117 Sum of reciprocal factorials by indefinite sums 121 Sums of exponential and trigonometric functions 123 Sums of other functions 129 Determination of sums by symbolical formulae 131 Determination of sums by generating functions 137 Determination of sums by geometrical considerations 138 Determination of sums by the Calculus of Probability 140 Stirlings Numbers numbers of the first kind 142 Determination of the Stirling numbers starting from their definition 145 Solution of the equations Smasm11 Som 147 Transformation of a multiple sum without repeti tion into sums without restriction 153 Stirlings numbers expressed by sums Limits 159 Application of the Stirling numbers of the first kind 163 Derivatives expressed by differences 164 Stirling numbers of the first kind obtained by proba bility 166 Stirling numbers of the second kind 168 Limits of expressions containing Stirling numbers of the second kind 173 Generating functions of the Stirling numbers of the second kind 174 Stirling numbers of the second kind obtained by probability 177 Decomposition of products of prime numbers into factors 179 Application of the expansion of powers into series of factorials 181 Formulae containing Stirling numbers of both kinds 182 Inversion of sums and series Sum equations 183 Deduction of certain formulae containing Stirling numbers 185 Differences expressed by derivatives 189 Expansion of a reciprocal factorial into a series of reciprocal powers and vice versa 192 The operation 195 The operation Y 199 Operations AMDm and Dm Am 200 Expansion of a function of function by aid of Stirling numbers Semiinvariants of Thiele 204 Expansion of a function into reciprocal factorial series and into reciprocal power series 212 Expansion of the function 1yn into a series of powers of x 216 Changing the origin 219 Changing the length of the interval 220 Stirlings polynomials 225 Bernoulli Polynomials and Numbers 78 Bernoulli polynomials of the first kind 230 Particular cases of the Bernoulli polynomials 236 Symmetry of the Bernoulli polynomials 238 Operations performed on the Bernoulli polynomial 240 Expansion of the Bernoulli polynomial into a Fourier series Limits Sum of reciprocal power series 242 Application of the Bernoulli polynomials 246 Expansion of a polynomial into Bernoulli polynomials 248 Expansion of functions into Bernoulli polynomials Generating functions 250 Raabes multiplication theorem for the Bernoulli polynomials 252 The Bernoulli series 253
 The Bernoulli series of the second kind 280 Gregorys summation formula 284 Eulers and Booles polynomials Sums of reciprocal powers 100 Eulers polynomials 288 Symmetry of the Euler polynomials 292 Expansion of the Euler polynomials into a series of Bernoulli polynomials of the first kind 295 Operations on the Euler polynomials 296 The Tangentcoefficients 298 Euler numbers 300 Limits of the Euler polynomials and numbers 302 Expansion of the Euler polynomials into Fourier series 303 Application of the Euler polynomials 306 Expansion of a polynomial into a series of Euler polynomials 307 Multiplication theorem of the Euler polynomials 311 Expansion of a function into an Euler series 313 Booles first summation formula 315 Booles polynomials 317 Operations on the Boole polynomials Differences 320 Expansion of the Boole polynomials into a series of Bernoulli polynomials of the second kind 321 Expansion of a function into Boole polynomials 322 Booles second summation formula 323 Sums of reciprocal powers Sum of 1x by aid of the digamma function 325 Sum of 1x2 by aid of the trigamma function 330 Sum of a rational fraction 335 Sum of reciprocal powers Sum of 1xm 338 Sum of alternate reciprocal powers by the B 347 Expansion of Functions Interpolation Construction of Tables 123 Expansion of a function into a series of polynomials 355 The Newton series 357 Interpolation by aid of Newtons formula and Construction of Tables 360 Inverse interpolation by Newtons formula 366 Interpolation by the Gauss series 368 The Bessel and the Stirling series 373 Everetts formula 376 Inverse interpolation by Everetts formula 381 Lagranges interpolation formula 385 Interpolation formula without printed differences 390 Inverse interpolation by aid of the formula of 411 Precision of the interpolation formulae 417 Examples of function chosen 434 Mathematical properties of the orthogonal poly 442 Approximation of a function given for 0 1 451 Computation of the binomial moments 460 Hermite polynomials 467 G polynomials 473 Numerical solution of equations Numerical 486 Method of iteration 492 Rootsquaring method Dandelin Lobatchevsky 503 Numerical integration 512 Hardy and Weddles formulae 516 The GaussLegendre method 517 Tchebichefs formula 519 Numerical integration of functions expanded into a series of their differences 524 Numerical solution of differential equations 527 Functions of several independent variables 161 Functions of two variables 530 Interpolation in a double entry table 532 Functions of three variables 541 Difference Equations 164 Genesis of difference equations 543 Homogeneous linear difference equations constant coefficients 545 Characteristic equations with multiple roots 549 Negative roots 552 Complex roots 554 Complete linear difference equations with con stant coefficients 557 Determination of the particular solution in the general case 564 Method of the arbitrary constants 569 Solution of linear difference equations by aid of generating functions 572 Homogeneous linear equations of the first order with variable coefficients 576 Laplaces method for solving linear homogeneous difference equations with variable coefficients 579 Complete linear difference equations of the first order with variable coefficients 583 Reducible linear difference equations with va riable coefficients 584 Linear difference equations whose coefficients are polynomials in x solved by the method of gen erating functions 586 Andrés method for solving difference equations 587 Sum equations which are reducible to equations 599 Solution of linear partial difference equations with 607 Booles symbolical method for solving partial dif 616 Homogeneous linear equations of mixed differences 632 Difference equations in four independent variables 638 Droits d'auteur