A Table of Generating Functions

1.1. Lagrange Series.

• The compendium applies Theorem 1.1 and a cited method to derive propositions and corresponding examples.The stated method uses Theorem 1.1 and the method in section (7).Source: §1.1 · p. 1

2. Introduction

• The introduction describes generating functions as tools for encoding sequences and addressing recurrence, partition, asymptotic, and probability questions.It also notes their use in coefficient-based counting and in encoding distributional information.Source: §2 · pp. 2–3

3.1. Entry 1.

• The supplied passages give explicit f(x) and φ(x) inputs in three cases, alongside fragments containing gamma factors, an Appell F1 expression, and a confluent hypergeometric 1F1 expression.Source: §3.1 · p. 3, §3.2 · p. 4, §3.4 · p. 5

3.8. Entry 9.

• Entries 9–11 give expressions involving gamma factors and confluent or ordinary hypergeometric functions.The listed forms include 1F1 and 2F1.Source: §3.8 · p. 7, §3.9 · p. 8, §3.10 · p. 8

3.12. Entry 13.

• Entries 12–15 present forms involving gamma factors, 2F1, and 3F2 functions.The supplied formula fragments include 3F2 in Entry 13 and 2F1 in Entry 15.Source: §3.12 · p. 9, §3.14 · p. 10

3.20. Entry 21.

• Entries 21–24 use 2F1 inputs or forms and include a gamma-factor expression in Entry 24.Entries 21–23 state f and φ; the Entry 24 fragments show gamma factors and a power of z.Source: The hypergeometric function in terms… · p. 12, §3.21 · p. 12, §3.22 · p. 13

3.26. Entry 27.

• Entry 27 states a proposition with a 2F1 expression, while the generalized Abel section includes a Gould-form example and a cited contour-integral method.The Gould form is compared with a similar form recorded in.Source: §3.26 · p. 14, §3.27 · p. 14

3.28. Entry 29.

• Entry 29 gives a 2F1 expression and an Appell F1 form, with f and φ specified for the example.The example is identified as a generating-function form in the supplied passages.Source: §3.28 · p. 15

4.3. Prudnikov equation (5.12.28).

• The section applies the method of section (7) in to equations in [14], and its examples include Example 4.7 from and Example 4.22, a Stirling-number case, from.Source: §4 · p. 16, §4.3 · pp. 17, 20

5. Extended Brychkov series.

• The Brychkov portion collects examples attributed to cited entries in, [13],,, and other listed sources.The supplied examples include an extension of Laplace’s first integral form and examples labeled Bell number and spherical Bessel function.Source: §5 · pp. 21, 23, 25, 30, 32

6. Definite integrals involving logarithm, quotient Pochammer and q-Pochammer functions

• The listed conditions require Re(a) < 0 and Re(m) < 0 for two examples, and Re(k) < 0 and Re(a) < 0 for another.Source: §6 · pp. 38–39
• Example 6.25 is identified as involving the logarithm of the q-Pochhammer symbol.Source: §6 · p. 38
• One example is attributed to equation (2.2.4.8) in.Source: §6 · p. 39

7. The Polylogarithm function

• The section develops a table involving the Hurwitz–Lerch zeta function and definite integrals of low-order polylogarithms.Source: §7 · p. 39
• The described derivation uses the binomial coefficient (n)j.Source: §7 · p. 39
• The derivation also uses equation (2.5.2.1) in and a method in.Source: §7 · p. 39

8. Logarithmic function generating functions

• One result fragment contains the factors (1 + 2p)! and (k −q)!.The fragment does not identify the full generating function or its closed form.Source: §8 · p. 42
• Examples 8.5–8.10 are attributed to equations in references –.Examples 8.5 and 8.6 cite and; the remaining entries cite or.Source: §8 · pp. 42–43
• The displayed conditions include Re(m) > 1, Re(b) > 1, and Re(σ) > −1/2.The same passage also gives a condition on Re(a), rendered as Re(a) > e3π.Source: §8 · p. 43

9. A few definite integrals

• Example 9.8 is attributed to Eq. (2.6.15.13) in.The accompanying result fragments include expressions involving log(ax) and log(log(ax)).Source: §9 · pp. 43–45
• Other displayed fragments repeat the term + 2i log(a) + i log.The log term's argument is absent from these fragments.Source: §9 · pp. 45–46
• Several result fragments contain terms involving log(a), log(b2+c2), and log(−2(π + i log(a) + i log(b))).These fragments do not preserve enough surrounding notation to identify the integral or full result.Source: §9 · pp. 44, 47

10. Extended Abel theorem forms

• One accompanying condition states that p is any positive integer.Other displayed conditions include Re(a) < 0 and Re(m) > 0, and a separate convergence condition on Re(a).Source: §10 · pp. 50, 55, 59
• The cited passages contain two 1F1 expressions and one 2F1 expression, with displayed parameters involving n, a, b, c, d, m, p, and z.Source: §10 · pp. 52–54
• Examples 10.22–10.25 draw on equations (1.9.54), (1.9.58), (1.9.54) and (1.9.55), and (1.14.5), respectively, in; Example 10.27 draws on Eq. (23) in.Source: §10 · pp. 54–55
• The section also includes result fragments with logarithmic factors and a 3F3 expression.One fragment contains 3F3(1, 1, 1; 2, 2, 2; m log(a)) log(a).Source: §10 · p. 75

11. Application of an Adams theorem

• The section says it applies Eq. (6.706) in to the formulae.This is the stated approach for the section's applications.Source: §11 · p. 77
• Example 11.10 is identified as the Bell number Bn.The passage names the quantity but does not preserve its full formula.Source: §11 · p. 79
• Examples 11.13, 11.27, 11.39, and 11.40 are drawn from equations in [13],,, and, respectively; their passages mention complete elliptic-integral and modified-Bessel-function material.Source: §11 · pp. 79, 82, 84
• Displayed result fragments include 1F1 and 2F1 expressions alongside Gamma factors.The fragments include Γ(1 + d + cn) 1F1(−n; 1 + d + (−1 + c)n; a + bn) and Γ(b + m + n) 2F1(−n, a(m + n); b + m; z).Source: §11 · p. 101

• The stated parameter conditions are Re(a) < 0 and Re(m) > 0.Source: §12 · p. 102

• A combinatorial approach derives the arcsin function in terms of the hypergeometric function.Source: §13 · p. 102
• The derivation yields a new special-function form for arcsin, described as not seen in current literature to the authors’ knowledge.Source: §13 · p. 102
• The new form allows the authors to write definite integrals involving elliptic functions and Catalan’s constant.Source: §13 · p. 102
• Arcsin is used in the special integral Owen T-function in wireless communications theory.Source: §13 · p. 102

• Examples 14.3–14.7 cite Eqs. (4.521.1), (4.521.3), (4.521.4), (4.521.6), and (4.521.7), respectively, from; Example 14.2 is introduced as Catalan’s constant.Source: §14 · pp. 103–104
• Examples 14.9–14.15 are identified with equations from [26], with Example 14.12 flagged as an error in the cited equation.Source: §14 · pp. 105–106
• Examples 14.16–14.21 are identified with equations from [26].Source: §14 · pp. 106–107
• Examples 14.22–14.25 are identified with equations from [13].Source: §14 · p. 108
• Examples 14.27–14.32 are identified as coming from Eqs. (2.6.1.1), (2.6.1.7), (2.6.1.18), (2.6.1.19), (2.6.1.20), and (2.6.1.21), respectively, in.Source: §14 · pp. 109–110
• Examples 14.36–14.39 and 14.41–14.46 are identified with equations from [6], while Example 14.40 cites.Source: §14 · pp. 112, 113, 114, 116