## Calculus of Finite Differences |

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### Contents

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 |

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 |

The Bernoulli series of the second kind | 280 |

Divided Differences | 18 |

Generating functions | 20 |

tangentcoefficient | 21 |

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 logTx+l 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 | 115 |

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 | 136 |

Determination of sums by geometrical considerations | 138 |

Determination of sums by the Calculus of Probability | 140 |

Stirlings Numbers 50 Expansion of factorials into power series Stirlings numbers of the first kind | 142 |

Determination of the Stirling numbers starting from their definition | 145 |

Solution of the equations SlS?lnS | 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 6 | 195 |

The operation | 199 |

Operations AmDm and DmAm | 200 |

Expansion of a function of function by aid of Stirling numbers Semiinvariants of Thiele | 204 |

arfx arithmetical mean of x 427 | 210 |

Expansion of a function into reciprocal factorial series and into reciprocal power series | 212 |

Expansion of the function 1y into a series of powers of x | 216 |

Changing the origin | 219 |

Changing the length of the interval | 220 |

Pni numhers | 221 |

Stirlings polynomials | 224 |

Bernoulli Polynomials and Numbers 78 Bernoulli polynomials of the first kind | 230 |

Bn Bernoulli numbers | 233 |

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 |

sum of reciprocal powers of degree m 1x | 244 |

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 MaclaurinEuler summation formula | 260 |

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 lx2 by aid of the trigamma function | 330 |

Sum of a rational fraction | 335 |

Sum of reciprocal powers Sum of 1x | 339 |

Sum of alternate reciprocal powers by the x | 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 | 358 |

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 |

Cmj Cotes numbers only in 131 and 155 | 388 |

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 |

Snm Stirling numhers of the first kind 142 | 448 |

Approximation of a function given for 1012 | 451 |

mo numbers approximation | 453 |

Computation of the binomial moments | 461 |

Hermite polynomials | 467 |

G polynomials | 473 |

D derivative 3 | 475 |

Numerical solution of equations Numerical | 486 |

Method of iteration | 492 |

The ChinVietaHorner method | 501 |

Rootsquaring method Dandelin Lobatchevsky | 511 |

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 | 533 |

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 |

Andres 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 |

649 | |

654 | |

### Common terms and phrases

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