Calculus of Finite Differences

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American Mathematical Soc., 1965 - 654 ページ

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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
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