Philippe Picard is a mathematician and probabilist, and was Professor at the Université de Lyon, France (retired in 2000), where he was responsible for the training of actuaries. His research focuses on mathematical tools in genetics, epidemics and risk theory, such as martingales and polynomials.Claude Lefèvre is a probabilist and statistician, and is Professor Emeritus at the Université Libre de Bruxelles, Belgium. His research focuses on applied probability models, in particular those related to epidemics, reliability, queueing and actuarial science.

• Preface ixChapter 1. Historical Abel-Gontcharoff Polynomials 11.1. Abel identity 11.2. Abel polynomials and expansions 21.3. Gontcharoff contribution 41.4. Increased recognition 81.5. A first meeting problem 101.6. A final epidemic outcome 131.7. A goodness-of-fit test 161.8. Extension to pseudopolynomials 19Chapter 2. Abel-Gontcharoff Pseudopolynomials 212.1. General framework: D, E, F,Δ 212.2. Copies Ε and standard families 242.3. An integration operator Iu 302.4. A-G pseudopolynomials Gn( |U) 322.5. Expansions of A–G type 372.6. A shift operator Sa 422.7. A multiplication operator Mλ 462.8. Shift invariance property 48Chapter 3. General Theory and Explicit Results 553.1. Return to the shift invariance 553.2. The higher dimensional case 663.3. Calculation formulas for ¯Gn( |U) 713.4. Geometric or affine form for U 803.5. Extension to special sequences ui = {ui,j} 873.6. When D is the set of integers 93Chapter 4. Further Results and Properties 974.1. A related basic family E(b) 974.2. Upper and lower bounds for Gn( |U) 1024.3. Short visit to the A–G type series 1114.4. Bilinear forms and biorthogonality 1164.5. An alternative generalization 119Chapter 5. Multi-index A–G Pseudopolynomials 1295.1. Key definitions and expansions 1295.2. Explicit formulas for Gn1,n2( |U) 1325.3. Multivariate case Gn1,n2( |U(1), U(2)) 1365.4. Integral multivariate representation 1465.5. Special case of A–G polynomials 149Chapter 6. Randomizing A-G Pseudopolynomials 1536.1. How to integrate stochasticity? 1536.2. With ui partial sums of i.i.d. variables 1556.3. Multivariate additive extension 1656.4. With ui partial products of i.i.d. variables 1706.5. Multivariate multiplicative extension 1736.6. Additive case for exponential functions 176Chapter 7. First Meeting Level with a Lower Boundary 1837.1. Return to a classical Poisson process 1837.2. For a compound Poisson process 1867.3. Related first passage problems 1907.4. With the number of Poisson jumps 1937.5. For a linear birth process with immigration 1967.6. Extension allowing multiple births 2017.7. For a nonlinear birth process 204Chapter 8. Less Standard First Meeting Models 2118.1. Compound Poisson process with a renewal process 2118.2. Linear birth process with a renewal process 2148.3. Nonlinear death process with a birth process 2178.4. Binomial process with a lower boundary 2218.5. For a compound binomial process 2248.6. Compound binomial process with a renewal process 226Chapter 9. Martingales and A–G Pseudopolynomials 2299.1. Motivation via damage-type models 2299.2. Unified treatment by A–G pseudopolynomials 2329.3. Reed–Frost multipopulation epidemic 2359.4. Nonlinear death process 2379.5. Combined general and fatal epidemics 2419.6. Time-dependent bivariate death process 246Chapter 10. Towards a Non-homogeneous Theory 25510.1. A non-stationary compound Poisson process 25510.2. A compound Poisson random field 265References 277Index 281

Philippe Picard is a mathematician and probabilist, and was Professor at the Université de Lyon, France (retired in 2000), where he was responsible for the training of actuaries. His research focuses on mathematical tools in genetics, epidemics and risk theory, such as martingales and polynomials.Claude Lefèvre is a probabilist and statistician, and is Professor Emeritus at the Université Libre de Bruxelles, Belgium. His research focuses on applied probability models, in particular those related to epidemics, reliability, queueing and actuarial science.

• Preface ixChapter 1. Historical Abel-Gontcharoff Polynomials 11.1. Abel identity 11.2. Abel polynomials and expansions 21.3. Gontcharoff contribution 41.4. Increased recognition 81.5. A first meeting problem 101.6. A final epidemic outcome 131.7. A goodness-of-fit test 161.8. Extension to pseudopolynomials 19Chapter 2. Abel-Gontcharoff Pseudopolynomials 212.1. General framework: D, E, F,Δ 212.2. Copies Ε and standard families 242.3. An integration operator Iu 302.4. A-G pseudopolynomials Gn( |U) 322.5. Expansions of A–G type 372.6. A shift operator Sa 422.7. A multiplication operator Mλ 462.8. Shift invariance property 48Chapter 3. General Theory and Explicit Results 553.1. Return to the shift invariance 553.2. The higher dimensional case 663.3. Calculation formulas for ¯Gn( |U) 713.4. Geometric or affine form for U 803.5. Extension to special sequences ui = {ui,j} 873.6. When D is the set of integers 93Chapter 4. Further Results and Properties 974.1. A related basic family E(b) 974.2. Upper and lower bounds for Gn( |U) 1024.3. Short visit to the A–G type series 1114.4. Bilinear forms and biorthogonality 1164.5. An alternative generalization 119Chapter 5. Multi-index A–G Pseudopolynomials 1295.1. Key definitions and expansions 1295.2. Explicit formulas for Gn1,n2( |U) 1325.3. Multivariate case Gn1,n2( |U(1), U(2)) 1365.4. Integral multivariate representation 1465.5. Special case of A–G polynomials 149Chapter 6. Randomizing A-G Pseudopolynomials 1536.1. How to integrate stochasticity? 1536.2. With ui partial sums of i.i.d. variables 1556.3. Multivariate additive extension 1656.4. With ui partial products of i.i.d. variables 1706.5. Multivariate multiplicative extension 1736.6. Additive case for exponential functions 176Chapter 7. First Meeting Level with a Lower Boundary 1837.1. Return to a classical Poisson process 1837.2. For a compound Poisson process 1867.3. Related first passage problems 1907.4. With the number of Poisson jumps 1937.5. For a linear birth process with immigration 1967.6. Extension allowing multiple births 2017.7. For a nonlinear birth process 204Chapter 8. Less Standard First Meeting Models 2118.1. Compound Poisson process with a renewal process 2118.2. Linear birth process with a renewal process 2148.3. Nonlinear death process with a birth process 2178.4. Binomial process with a lower boundary 2218.5. For a compound binomial process 2248.6. Compound binomial process with a renewal process 226Chapter 9. Martingales and A–G Pseudopolynomials 2299.1. Motivation via damage-type models 2299.2. Unified treatment by A–G pseudopolynomials 2329.3. Reed–Frost multipopulation epidemic 2359.4. Nonlinear death process 2379.5. Combined general and fatal epidemics 2419.6. Time-dependent bivariate death process 246Chapter 10. Towards a Non-homogeneous Theory 25510.1. A non-stationary compound Poisson process 25510.2. A compound Poisson random field 265References 277Index 281

Philippe Picard is a mathematician and probabilist, and was Professor at the Université de Lyon, France (retired in 2000), where he was responsible for the training of actuaries. His research focuses on mathematical tools in genetics, epidemics and risk theory, such as martingales and polynomials.Claude Lefèvre is a probabilist and statistician, and is Professor Emeritus at the Université Libre de Bruxelles, Belgium. His research focuses on applied probability models, in particular those related to epidemics, reliability, queueing and actuarial science.

• Preface ixChapter 1. Historical Abel-Gontcharoff Polynomials 11.1. Abel identity 11.2. Abel polynomials and expansions 21.3. Gontcharoff contribution 41.4. Increased recognition 81.5. A first meeting problem 101.6. A final epidemic outcome 131.7. A goodness-of-fit test 161.8. Extension to pseudopolynomials 19Chapter 2. Abel-Gontcharoff Pseudopolynomials 212.1. General framework: D, E, F,Δ 212.2. Copies Ε and standard families 242.3. An integration operator Iu 302.4. A-G pseudopolynomials Gn( |U) 322.5. Expansions of A–G type 372.6. A shift operator Sa 422.7. A multiplication operator Mλ 462.8. Shift invariance property 48Chapter 3. General Theory and Explicit Results 553.1. Return to the shift invariance 553.2. The higher dimensional case 663.3. Calculation formulas for ¯Gn( |U) 713.4. Geometric or affine form for U 803.5. Extension to special sequences ui = {ui,j} 873.6. When D is the set of integers 93Chapter 4. Further Results and Properties 974.1. A related basic family E(b) 974.2. Upper and lower bounds for Gn( |U) 1024.3. Short visit to the A–G type series 1114.4. Bilinear forms and biorthogonality 1164.5. An alternative generalization 119Chapter 5. Multi-index A–G Pseudopolynomials 1295.1. Key definitions and expansions 1295.2. Explicit formulas for Gn1,n2( |U) 1325.3. Multivariate case Gn1,n2( |U(1), U(2)) 1365.4. Integral multivariate representation 1465.5. Special case of A–G polynomials 149Chapter 6. Randomizing A-G Pseudopolynomials 1536.1. How to integrate stochasticity? 1536.2. With ui partial sums of i.i.d. variables 1556.3. Multivariate additive extension 1656.4. With ui partial products of i.i.d. variables 1706.5. Multivariate multiplicative extension 1736.6. Additive case for exponential functions 176Chapter 7. First Meeting Level with a Lower Boundary 1837.1. Return to a classical Poisson process 1837.2. For a compound Poisson process 1867.3. Related first passage problems 1907.4. With the number of Poisson jumps 1937.5. For a linear birth process with immigration 1967.6. Extension allowing multiple births 2017.7. For a nonlinear birth process 204Chapter 8. Less Standard First Meeting Models 2118.1. Compound Poisson process with a renewal process 2118.2. Linear birth process with a renewal process 2148.3. Nonlinear death process with a birth process 2178.4. Binomial process with a lower boundary 2218.5. For a compound binomial process 2248.6. Compound binomial process with a renewal process 226Chapter 9. Martingales and A–G Pseudopolynomials 2299.1. Motivation via damage-type models 2299.2. Unified treatment by A–G pseudopolynomials 2329.3. Reed–Frost multipopulation epidemic 2359.4. Nonlinear death process 2379.5. Combined general and fatal epidemics 2419.6. Time-dependent bivariate death process 246Chapter 10. Towards a Non-homogeneous Theory 25510.1. A non-stationary compound Poisson process 25510.2. A compound Poisson random field 265References 277Index 281