Combinatorial numbers and Egorychev method

These are binomial coefficients, Stirling numbers, Catalan numbers, Harmonic numbers, Eulerian numbers and Bernoulli numbers, as in the book Integral representation and the Computation of Combinatorial Sums by G. P. Egorychev.

The collection including a list of all identities proved is available here (part 1 and 2, formal power series and residue operators):

and here (part 3, complex variables) This includes a number of identities from H.W. Gould's book Combinatorial identities as well as L. Saalschütz's book on Bernoulli numbers.

We now have a new section as of March 2023 "computer search" which includes hypergeometric identities that can prove challenging even for computer algebra systems, such as

As of January 2024 "computer search" continues with rare identities such as (the 42 identity)

also featuring Narayana and Catalan numbers. These computer identities have ranges of validity (boundary conditions) which are documented in the text.

Here are the slides from Hosam Mahmoud's talk on Egorychev method at Catholic University on November 9, 2022: History and examples of Egorychev method.

The 2023 paper "Egorychev method: a hidden treasure" by Riedel and Mahmoud is at the following Springer link.

My home page is here.

A MathJax version of this list can be found here.

This is the list of identities in the above document, including links to the posts where they first appeared.

Part 1 and 2

Egorychev method in formal power series

  1. MSE 2384932

  2. MSE 2472978

  3. MSE 2719320

  4. MSE 2830860

  5. MSE 2904333

  6. MSE 2950043

  7. MSE 3049572

  8. MSE 3051713

  9. MSE 3068381

  10. MSE 3138710

  11. MSE 3196998

  12. MSE 3245099

  13. MSE 3260307

  14. MSE 3285142

  15. MSE 3333597

  16. MSE 3342361

  17. MSE 3383557

  18. MSE 3441855

  19. MSE 3577193

  20. MSE 3583191

  21. MSE 3592240

  22. MSE 3604802

  23. MSE 3619182

  24. MSE 3638162

  25. MSE 3661349

  26. MSE 3706767

  27. MSE 3737197

  28. MSE 3825092

  29. MSE 3845061

  30. MSE 3885278

  31. MSE 3559223

  32. MSE 3926409

  33. MSE 3942039

  34. MSE 3956698

  35. MSE 3993530

  36. MSE 4008277

  37. MSE 4031272 (Li Shanlan identity)

  38. MSE 4034224 (Eulerian numbers, Stirling numbers first and second kind)

  39. MSE 4037172 (Eulerian numbers, associated Stirling numbers first and second kind)

  40. MSE 4037946 (Eulerian numbers, associated Stirling numbers first and second kind)

  41. MSE 4055292

  42. MSE 4054024

  43. MSE 4084763

  44. MSE 4095795

  45. MSE 4098492

  46. MSE 4127695

  47. MSE 4131219

  48. MSE 4139722

  49. MSE 4192271

  50. MSE 4212878

  51. A different obstacle from Concrete Mathematics by Knuth, Graham, and Patashnik

  52. MO 291738 (not quite the same as previous)

  53. A Stirling number identity by Gould

  54. A Stirling number identity by Gould II

  55. Schläfli's identity for Stirling numbers

  56. Stirling numbers and Faulhaber's formula

  57. Stirling numbers and binomial coefficient

  58. Stirling numbers and a double binomial coefficient

  59. Stirling numbers and a double binomial coefficient II

  60. Stirling numbers and Bernoulli polynomials

  61. Central binomial coefficients and Stirling numbers

  62. Single variable monomial and two binomial coefficients

  63. Use of an Iverson bracket

  64. Use of an Iverson bracket II

  65. Use of an Iverson bracket III

  66. Basic example

  67. Basic example continued

  68. An identity by Erik Sparre Andersen

  69. Very basic example

  70. An identity by Karl Goldberg

  71. Sum producing a square root

  72. Sum producing a square root II

  73. Use of an Iverson bracket IV

  74. Binomial coefficient manipulation

  75. Four binomial sums

  76. Power term and two binomial coefficients

  77. Use of an Iverson bracket V

  78. Use of an Iverson bracket VI

  79. Appearance of the constants three and five

  80. Generating function of a binomial term

  81. Double square root

  82. Central Delannoy numbers

  83. A case of factorization

  84. Two identities due to Grosswald

  85. Appearance of the constant three

  86. Very basic example

  87. Very basic example II

  88. Nested square root

  89. Harmonic numbers and a squared binomial coefficient

  90. Harmonic numbers and a double binomial coefficient

  91. Two instances of a Harmonic number

  92. Legendre Polynomials

  93. Legendre Polynomials and a square root

  94. Legendre Polynomials and a double square root

  95. MSE 4304623

  96. Legendre Polynomials, trigonometric terms and a contour integral

  97. Sum independent of a variable

  98. Polynomial in three variables

  99. An identity by Van der Corput

  100. MSE 4316037 Logarithm, binomial coefficient and harmonic numbers

  101. An identity credited to Chung

  102. MSE 4317353 A Catalan number convolution

  103. Odd index binomial coefficients

  104. MSE 4325592 A sum of inverse binomial coefficients

  105. Inverted sum index

  106. MSE 4351714 A Catalan number recurrence

  107. An identity by Graham and Riordan

  108. Square root term

  109. An identity by Machover and Gould

  110. Moriarty identity by H.T.Davis et al.

  111. Inverse Moriarty identity by Marcia Ascher.

  112. Moriarty identity by Egorychev.

  113. MSE 4462359 Two binomial coefficients

  114. Polynomial identity

  115. Polynomial identity II

  116. Worpitzky-Nielsen series

  117. MSE 4517120 A sum of inverse binomial coefficients

  118. MSE 4520057 Symmetric Bernoulli number identity

  119. Polynomial identity III

  120. Polynomial identity IV

  121. MSE 4540192 Symmetry in a simple proof

  122. Free functional term

  123. MSE 4547110 Inverse central binomial coefficient

  124. MSE 1402886 Inverse binomial coefficient

  125. MSE 4552694 A pair of binomial transforms

  126. Polynomial with inverse binomial coefficients

  127. Harmonic numbers with inverse binomial coefficients

  128. Simon's identity

  129. Identity from Abramowitz and Stegun / Schläfli's formula

  130. Bernoulli / Stirling number identity identity

  131. Recurrence relation from DLMF

  132. Bernoulli, Fibonacci and Lucas numbers

  133. Bernoulli / exponential convolution

  134. Bernoulli identity by Munch

  135. Bernoulli identity by Kronecker

  136. Computing Bernoulli numbers

From the Saalschütz text

  1. Worpitzky's identity

  2. MSE 4627726: Quadruple binomial coefficient

  3. MSE 4627918: Alternating power sum

  4. MSE 4227433: Squared power sum

  5. MSE 4428892: Ordinary power sum

  6. MSE 3932757: Stirling numbers and a tree-function like term

  7. MSE 4641290: A vanishing variable

  8. MSE 4644963: From trigonometric to rational

  9. MSE 4657112: Triple combinatorial numbers to constant

Computer search

  1. MSE 4667102: Two different representations of a coefficient

  2. MSE 4675665: Rational term of constant degree

  3. MSE 4666141: Double square root

  4. MSE 4699857: Four auxiliary variables

  5. MSE 4703564: A family of odd polynomials

  6. MSE 4713851: A sum with a zero value

  7. MSE 4722503: Euler numbers and Stirling numbers

  8. MSE 495371: Even-index binomial coefficient convolution

  9. MSE 4731417: Kravchuk polynomials

  10. MSE 4762542: Binomial-Bernoulli convolution

  11. MSE 4774167 : Two probabilities

  12. MSE 4791957 : Motzkin numbers

  13. MSE 4821034 : An inverse binomial coefficient

  14. MSE 4830342 : Euler numbers, Stirling numbers and Touchard polynomials

  15. MSE 4832009 : A triple binomial

  16. MSE 4843051 : Double sum with an absolute value

  17. MSE 4850609 :Inverse central binomial coefficient in sum

Computer search II

  1. MSE 4902676: Inverse central binomial coefficient

  2. MSE 4906245: Another inverse binomial coefficient

  3. Quadruple hypergeometric with an inverse binomial coefficient

  4. Quadruple hypergeometric with two inverse binomial coefficients

  5. MSE 4949815: Inverse binomial coefficient with harmonic numbers

  6. MSE 4960823: Binomial coefficient scaled power sum

  7. MSE 4964157: Two central binomial coefficients and a fractional term

Part 3

Egorychev method in complex variables

  1. Introductory example for the method

  2. Introductory example for the method, convergence about zero

  3. Introductory example for the method, an interesting substitution

  4. Introductory example for the method, another interesting substitution

  5. Introductory example for the method, yet another interesting substitution

  6. Using an Iverson bracket only

  7. Verifying that a certain sum vanishes

  8. A case of radical cancellation

  9. Basic usage of exponentiation integral

  10. Introductory example for the method, eliminating odd-even dependence

  11. Introductory example for the method, proving equality of two double hypergeometrics

  12. A remarkable case of factorization

  13. Evaluating a quadruple hypergeometric

  14. An integral representation of a binomial coefficient involving the floor function

  15. Evaluating another quadruple hypergeometric

  16. An identity by Strehl

  17. Shifting the index variable and applying Leibniz' rule

  18. Working with negative indices

  19. Two companion identities by Gould

  20. Exercise 1.3 from Stanley's Enumerative Combinatorics

  21. Counting m-subsets

  22. Method applied to an iterated sum

  23. A pair of two double hypergeometrics

  24. A two phase application of the method

  25. An identity from Mathematical Reflections

  26. A triple Fibonacci-binomial coefficient convolution

  27. Fibonacci numbers and the residue at infinity

  28. Permutations containing a given subsequence

  29. An example of Lagrange inversion

  30. A binomial coefficient - Catalan number convolution

  31. A new obstacle from Concrete Mathematics

  32. Abel-Aigner identity from Table 202 of Concrete Mathematics

  33. Reducing the form of a double hypergeometric

  34. Basic usage of the Iverson bracket

  35. Basic usage of the Iverson bracket II

  36. Use of a double Iverson bracket

  37. Iverson bracket and an identity by Gosper, generalized

  38. Factoring a triple hypergeometric sum

  39. Factoring a triple hypergeometric sum II

  40. Factoring a triple hypergeometric sum III

  41. A triple hypergeometric sum IV

  42. Basic usage of exponentiation integral to obtain Stirling number formulae

  43. Three phase application including Leibniz' rule

  44. Symmetry of the Euler-Frobenius coefficient

  45. A probability distribution with two parameters

  46. An identity involving Narayana numbers

  47. Convolution of Narayana polynomials

  48. A property of Legendre polynomials

  49. A sum of factorials, OGF and EGF of the Stirling numbers of the second kind

  50. Fibonacci, Tribonacci, Tetranacci

  51. Stirling numbers of two kinds, binomial coefficients

  52. An identity involving involving two binomial coefficients and a fractional term

  53. Double chain of a total of three integrals

  54. Rothe-Hagen identity

  55. Abel polynomials are of binomial type

  56. A summation identity with four poles

  57. A summation identity over odd indices with a branch cut

  58. A Stirling-number identity

  59. A Catalan-Central binomial coefficient convolution

Post Scriptum section

  1. A trigonometric sum

  2. A class of polynomials similar to Fibonacci and Lucas Polynomials

  3. Partial row sums in Pascal's triangle

  4. The tree function and Eulerian numbers of the second order

  5. A Stirling set number generating function and Eulerian numbers of the second order

    A Stirling cycle number generating function and Eulerian numbers of the second order (II)

  6. Another case of factorization

  7. An additional case of factorization

  8. Contours and a binomial square root

  9. A careful examination of contours

  10. Stirling numbers, Bernoulli numbers and Catalan numbers from Concrete Mathematics by Graham, Knuth and Patashnik

  11. Transforming an OGF into an EGF

  12. Stirling numbers of the first and second kind

  13. An identity by Carlitz

  14. Logarithm of the Catalan number OGF

  15. A Bernoulli / Stirling number identity

  16. Formal power series vs contour integration

Computer search III