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5. Examples

  • Francesco Aldo Costabile
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Modern Umbral Calculus
This chapter is in the book Modern Umbral Calculus

Abstract

In this chapter, we will discuss some important examples of b. p. s. without any pretence of doing a complete treatise on b. p. s. We will follow this order: 1. Numerical sequences: b := (bn)n∈ℕ and b̂:= (̂b̂n)n∈ℕ; 2. Exponential Riordan matrices associated: P and ̂P; 3. Polynomial sequences {pn}n and {̂pn}n; 4. Generating function, binomial identity, recurrence and differential equations, determinantal forms; 5. δ-functional and δ-operator associated, functional equation; 6. Applications to interpolation and approximation. Naturally the binomial identity, the functional equation, et cetera, will be repeated only if they are explicit or significant.

Abstract

In this chapter, we will discuss some important examples of b. p. s. without any pretence of doing a complete treatise on b. p. s. We will follow this order: 1. Numerical sequences: b := (bn)n∈ℕ and b̂:= (̂b̂n)n∈ℕ; 2. Exponential Riordan matrices associated: P and ̂P; 3. Polynomial sequences {pn}n and {̂pn}n; 4. Generating function, binomial identity, recurrence and differential equations, determinantal forms; 5. δ-functional and δ-operator associated, functional equation; 6. Applications to interpolation and approximation. Naturally the binomial identity, the functional equation, et cetera, will be repeated only if they are explicit or significant.

Chapters in this book

  1. Frontmatter I
  2. Preface VII
  3. Acknowledgment IX
  4. Contents XI
  5. Acronyms XV
  6. Part I: Introduction
  7. 1. Preliminaries and notations 3
  8. 2. Particular matrices and their connections with formal power series 7
  9. Part II: Polynomial sequences of binomial type
  10. 3. Binomial polynomial sequences 23
  11. 4. Applications to linear interpolation and operators approximation theory 41
  12. 5. Examples 47
  13. Part III: Appell polynomial sequences
  14. 6. Appell polynomial sequences 77
  15. 7. Application to linear interpolation and approximation theory 101
  16. 8. Examples 107
  17. Part IV: Sheffer polynomial sequences
  18. 9. Sheffer polynomial sequence 149
  19. 10. Applications to linear interpolation and operators approximation theory 167
  20. 11. Examples 175
  21. Part V: Lidstone polynomial sequences
  22. 12. Lidstone-type polynomial sequences 203
  23. 13. Application to linear interpolation and operators approximation theory 217
  24. 14. Examples 223
  25. Bibliography 245
  26. Index 255
https://doi.org/10.1515/9783110652925-008

Chapters in this book

  1. Frontmatter I
  2. Preface VII
  3. Acknowledgment IX
  4. Contents XI
  5. Acronyms XV
  6. Part I: Introduction
  7. 1. Preliminaries and notations 3
  8. 2. Particular matrices and their connections with formal power series 7
  9. Part II: Polynomial sequences of binomial type
  10. 3. Binomial polynomial sequences 23
  11. 4. Applications to linear interpolation and operators approximation theory 41
  12. 5. Examples 47
  13. Part III: Appell polynomial sequences
  14. 6. Appell polynomial sequences 77
  15. 7. Application to linear interpolation and approximation theory 101
  16. 8. Examples 107
  17. Part IV: Sheffer polynomial sequences
  18. 9. Sheffer polynomial sequence 149
  19. 10. Applications to linear interpolation and operators approximation theory 167
  20. 11. Examples 175
  21. Part V: Lidstone polynomial sequences
  22. 12. Lidstone-type polynomial sequences 203
  23. 13. Application to linear interpolation and operators approximation theory 217
  24. 14. Examples 223
  25. Bibliography 245
  26. Index 255
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