### 目次

 On Operations 1 Symbolical Calculus 7 Central Differences 15 5 22 Operations on the Euler polynomials 23 Beta functions 24 Incomplete Beta functions 25 Exponential functions 26
 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 Stirlings polynomials 225 Operations performed on the Bernoulli polynomial 240 Application of the Bernoulli polynomials 247 The Bernoulli series 253

 Trigonometric functions 27 Alternate functions 28 Expansion of functions into power series 29 Product of two functions Differences 30 Product of two functions Means 31 The Gamma function 53 Bernoulli polynomials of the first kind 78 94 94 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 37 Differences sums and means of infinite series 38 Inverse operation of the mean 109 39 Other methods of obtaining inverse means 40 Sums 41 Sums determined by indefinite sums 43 Sums of exponential and trigonometric functions 123 Sums of other functions 129 The Trigamma function 130 Determination of sums by symbolical formulae 131 Interpolation formula without printed differences 132 Determination of sums by generating functions 136 Determination of sums by geometrical considerations 138 Determination of sums by the Calculus of Probability 49 Determination of alternate sums starting from usual sums Chapter IV Stirlings Numbers 140 140 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 S Sm nSm 147 Transformation of a multiple sum without repeti tion into sums without restriction 153 Numerical integration 155 Hardy and Weddles formulae 156 The GaussLegendre method 157 Tchebichefs formula 158 Stirlings numbers expressed by sums Limits 159 Numerical solution of differential equations 160 Functions of two variables 161 Interpolation in a double entry table 162 Application of the Stirling numbers of the first kind 163 Derivatives expressed by differences 57 Stirling numbers of the first kind obtained by proba bility 153 159 163 164 the second kind 173 60 Generating functions of the Stirling numbers of the second kind 173 Expansion of log x+1 into a power series 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 179 181 182 183 69 The operation 70 The operation 71 Opera... 192 72 Expansion of a function of function by aid of Stirling numbers Semiinvariants of Thiele 204
 The MaclaurinEuler summation formula 260 Symmetry of the Bernoulli polynomials of the second 268 Operations on the Bernoulli polynomials of 275 Gregorys summation formula 284 Application of the Euler polynomials 306 Expansion of a function into an Euler series 313 Operations on the Boole polynomials Differences 320 Expansion of Functions Interpolation 355 Inverse interpolation by Newtons formula 366 The Bessel and the Stirling series 373 Inverse interpolation by Everetts formula 381 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 503 530 532 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 general case 170 Determination of the particular solution in 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 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 638 276 648 erating functions 564 649 著作権