Calculus of Finite Differences

Front Cover
American Mathematical Soc., Jan 1, 1965 - Mathematics - 654 pages
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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
8 central difference 15
649
F x trigamma function
654
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