Faculty series

The concept of faculty series ( English factorial series ) comes from the mathematics . The faculty series are part of the function series and are closely related to the Dirichlet series . Last but not least, they are of particular importance when studying difference equations .

definition

If a sequence of real or complex numbers is given, the series of functions is ${\displaystyle {\mathcal {A}}=\left(a_{n}\right)_{n=0,1,2,3,\ldots }}$

${\displaystyle {\begin{aligned}\Omega (z)&=\sum _{n=0}^{\infty }{\frac {n!a_{n}}{z(z+1)\cdots (z+n)}}\\&={\frac {a_{0}}{z}}+{\frac {a_{1}}{z(z+1)}}+{\frac {2a_{2}}{z(z+1)(z+2)}}+{\frac {6a_{3}}{z(z+1)(z+2)(z+3)}}+\cdots +{\frac {n!a_{n}}{z(z+1)\cdots (z+n)}}+\cdots \end{aligned}}}$

with the series of faculties ( belonging to the series ) . ${\displaystyle z\in \mathbb {C} \setminus \{0,-1,-2,\ldots \}}$ ${\displaystyle {\mathcal {A}}}$

Some authors assume that is the initial term . Other authors, on the other hand, even allow a real or complex constant to be added to the above and also refer to the series of functions thus given as a factorial series. All of these views of the concept of the faculty series are essentially equivalent. ${\displaystyle a_{0}=0}$ ${\displaystyle \Omega (z)}$ ${\displaystyle c}$ ${\displaystyle c+\Omega (z)}$

Convergence behavior

A few sentences provide information about the convergence behavior of the faculty series , which can be traced back not least to mathematicians such as Edmund Landau , Johan Ludwig Jensen , Salvatore Pincherle and Niels Erik Nørlund .

Landau's theorem

This sentence found by Edmund Landau brings the question of the convergence behavior of the faculty series in connection with the corresponding question for the Dirichlet series. It says namely:

The faculty series described above and the associated Dirichlet series ${\displaystyle \Omega (z)}$

${\displaystyle \Psi (z)=\sum _{n=1}^{\infty }{\frac {a_{n}}{n^{z}}}}$

have the same convergence behavior within the area . The following applies in detail: ${\displaystyle \mathbb {C} \setminus \{0,-1,-2,\ldots \}}$

(I) The two series and are convergent and divergent for one and the same . ${\displaystyle \Omega (z)}$ ${\displaystyle \Psi (z)}$ ${\displaystyle z\in \mathbb {C} \setminus \{0,-1,-2,\ldots \}}$

(II) If uniformly convergent on a closed circular disk , then this also applies for and only then. ${\displaystyle \Omega (z)}$ ${\displaystyle {\overline {U_{r}(z_{0})}}\subset \mathbb {C} \setminus \{0,-1,-2,\ldots \}\;\;(r>0\;,\;z_{0}\neq 0,-1,-2,\ldots )}$ ${\displaystyle \Psi (z)}$

Jensen's and Pincherle's sentences

Jensen's theorem deals with the question of the nature of the convergence domain of the faculty series. It says the following:

At a faculty series there is always a - as Konvergenzabszisse called - finite or infinite number such that for every complex number of the area diverges and for every complex number of the area converges. The convergence area of a factorial series is thus a half-plane open to the right , from which (if necessary) the zero and the negative integers have been removed. ${\displaystyle \Omega (z)}$ ${\displaystyle \lambda \in \mathbb {R} \cup \{-\infty ,+\infty \}}$ ${\displaystyle \Omega (z)}$ ${\displaystyle \{z\in \mathbb {C} \setminus \{0,-1,-2,\ldots \}\colon \operatorname {Re} (z)<\lambda \}}$ ${\displaystyle \{z\in \mathbb {C} \setminus \{0,-1,-2,\ldots \}\colon \operatorname {Re} (z)>\lambda \}}$

Pincherle's theorem deals with the corresponding question with regard to the absolute convergence of the faculty series and can be stated as follows:

The area of ​​the absolute convergence of a factorial series is also a half-plane open to the right, from which the zero and the negative integers (if applicable) have been removed. For a series of faculties there is always a finite or infinite number - also called the abscissa of absolute convergence - such that it is absolutely convergent in the area . For every complex number, with converges, but not absolutely converges. The width of the infinite strip between the two abscissas is at most ; so the inequality applies . ${\displaystyle \Omega (z)}$ ${\displaystyle \mu \in \mathbb {R} \cup \{-\infty ,+\infty \}}$ ${\displaystyle \Omega (z)}$ ${\displaystyle \{z\in \mathbb {C} \setminus \{0,-1,-2,\ldots \}\colon \operatorname {Re} (z)>\mu \}}$ ${\displaystyle \Omega (z)}$ ${\displaystyle z}$ ${\displaystyle \lambda <\operatorname {Re} (z)<\mu }$ ${\displaystyle 1}$ ${\displaystyle \lambda \leq \mu \leq \lambda +1}$

Nørlund's theorem

Niels Erik Nørlund found this sentence and thus clarified the question of the even convergence of series of faculties. The sentence can be formulated as follows:

Let the series of faculties converge at a point . Furthermore, an arbitrary positive number is given and in addition the angular field anchored in the point , opened to the right , the two legs of which emerge from the two half-straight lines extending from the point perpendicular to the real axis through two - radian - rotations . ${\displaystyle \Omega (z)}$ ${\displaystyle z_{0}\in {\mathbb {C} \setminus \{0,-1,-2,\ldots \}}}$ ${\displaystyle \eta <{\frac {\pi }{2}}}$ ${\displaystyle z_{0}}$ ${\displaystyle W(z_{0},\eta )\subset \mathbb {C} \setminus \{0,-1,-2,\ldots \}}$ ${\displaystyle \eta }$ ${\displaystyle z_{0}}$

Then:

${\displaystyle \Omega (z)}$is always uniformly convergent on the angular field . ${\displaystyle W(z_{0},\eta )}$

analogue

As GM Fichtenholz explains in his differential and integral calculus II , - according to a theorem by Konrad Knopp - with regard to the convergence behavior, the relationships between a faculty series and its Dirichlet series are similar to those that exist between a Lambert series and the power series belonging to this Lambert series .

Holomorphism

The complex functions given by the factorial series have - in the same way as the complex functions given by the associated Dirichlet series - some regularity properties. This is based on a combination of Weierstrasse's convergence theorem with Nørlund's theorem. Overall, the following sentence applies:

For a factorial series as given above, a holomorphic function is defined by the assignment on the convergence half- plane. This function (also labeled) has the following derivation function : ${\displaystyle z\mapsto \Omega (z)}$ ${\displaystyle \{z\in \mathbb {C} \setminus \{0,-1,-2,\ldots \}\colon \operatorname {Re} (z)>\lambda \}}$ ${\displaystyle \Omega (z)}$

${\displaystyle \Omega ^{'}(z)=-\sum _{n=1}^{\infty }{\frac {n!\cdot \left({\frac {a_{0}}{n}}+{\frac {a_{1}}{n-1}}+\cdots +{\frac {a_{n-1}}{1}}\right)}{z(z+1)\cdots (z+n)}}}$  .

Further representations

Faculty series can also be represented with the help of the gamma function and Euler's beta function. The following applies:

A factorial series as given above always satisfies the equations :

${\displaystyle {\begin{aligned}\Omega (z)&=\Gamma (z)\cdot \sum _{n=0}^{\infty }{\frac {n!a_{n}}{\Gamma (z+n+1)}}\\&=\sum _{n=0}^{\infty }{a_{n}\cdot \mathrm {B} (z+n+1)}\;\;(z\in \mathbb {C} \setminus \{0,-1,-2,\ldots \})\\\end{aligned}}}$

literature

• GM Fichtenholz: Differential and Integral Calculus II . Translation from Russian and scientific editing: Dipl.-Math. Brigitte Mai, Dipl.-Math. Walter Mai (=  university books for mathematics . Volume 62 ). 6th edition. VEB Deutscher Verlag der Wissenschaften , Berlin 1974.
• Konrad Knopp: Theory and Application of the Infinite Series (=  The Basic Teachings of Mathematical Sciences . Volume 2 ). 5th, corrected edition. Springer Verlag , Berlin, Göttingen, Heidelberg, New York 1964, ISBN 3-540-03138-3 ( MR0183997 ).
• Edmund Landau: About the basics of the theory of the faculty series . In: Meeting reports of the Bavarian Academy of Sciences . tape 36 , 1906, pp. 151-218 .
• Edmund Landau: Comment on my essay: On the basics of the theory of the faculty series . In: Meeting reports of the Bavarian Academy of Sciences . tape 39 , 1909, pp. 7-10 .
• LM Milne-Thomson : The Calculus of Finite Differences . Re-issued by arrangemant with The Macmillan Press. 2nd, unchanged edition. Chelsea Publishing Company , New York 1981, ISBN 0-8284-0308-2 , X ( MR0185152 ).
• Niels Nielsen : The gamma function . Volume I: Handbook of the theory of the gamma function. Third part: Theory of the faculty series. Chelsea Publishing Company, Bronx, New York 1965, XVII ( MR0043339 ).
• Niels Erik Nörlund : Lectures on calculus of differences (=  The Basic Teachings of Mathematical Sciences . Volume 13 ). Springer Verlag , Berlin 1924, ISBN 978-3-642-50514-0 , chap. 9 , doi : 10.1007 / 978-3-642-50824-0_10 ( MR1549596 ).

Individual evidence

1. ↑ E. Landau: About the basics of the theory of the faculty series. Meeting reports of the Bavarian Academy of Sciences 36, p. 151 ff
2. ^ LM Milne-Thomson: The Calculus of Finite Differences. 1981, p. 271 ff
3. ↑ ^{a b} Niels Nielsen: Handbook of the theory of the gamma function. Chapter XVII 1965, p. 237 ff
4. ↑ ^{a b} G. M. Fichtenholz: differential and integral calculus II. 1974, p. 322
5. ↑ Konrad Knopp: Theory and application of the infinite series. 1964, p. 462 ff
6. ↑ Niels Erik Nörlund: Lectures on calculus of differences. 1924, p. 256 ff
7. ↑ E. Landau: About the basics of the theory of the faculty series. Meeting reports of the Bavarian Academy of Sciences 36, p. 167
8. ↑ Knopp, op.cit., P. 462
9. ↑ ^{a b} Nielsen, op.cit., P. 245
10. ↑ LM Milne-Thomson, op.cit., Pp. 275 ff
11. ↑ ( English abscissa of convergence )
12. ↑ ( English region of convergence )
13. ↑ In the borderline case , the convergence region is the empty set . Nevertheless, the term area is also used here . In the same way one speaks in the other borderline case , although here the convergence area is the entire area , i.e. an infinitely often dotted plane , also of a half plane. ${\displaystyle \lambda =+\infty }$ ${\displaystyle \lambda =-\infty }$ ${\displaystyle \{z\in \mathbb {C} \setminus \{0,-1,-2,\ldots \}\colon \operatorname {Re} (z)<\lambda \}}$
14. ↑ Milne-Thomson, op.cit., P. 276
15. ↑ ( English abscissa of absolute convergence )
16. ↑ Nörlund, op.cit., P. 258
17. ↑ LM Milne-Thomson, op. Cit., Pp. 284-287
18. ↑ The angles are given here in radians . The point is both the center of rotation of the two rotations and the vertex of the angle determined by the angular field, which is. In the two rotations, the lower half-line is transferred into the lower leg of the angular field by rotating around the angle in the mathematically positive direction of rotation and the upper half-line is transferred to the upper leg by rotating the angle in the mathematically negative direction of rotation. ${\displaystyle z_{0}}$ ${\displaystyle (\pi -2\eta ){\text{ }}\mathrm {rad} }$ ${\displaystyle z_{0}-{\mathrm {i} }\cdot \mathbb {R} ^{\geq 0}}$ ${\displaystyle \eta {\text{ }}\mathrm {rad} }$ ${\displaystyle z_{0}+{\mathrm {i} }\cdot \mathbb {R} ^{\geq 0}}$ ${\displaystyle -\eta {\text{ }}\mathrm {rad} }$
19. ↑ Fichtenholz, op.cit., P. 323
20. ↑ Nørlund's theorem implies that a faculty series converges locally uniformly in every point of its convergence area .
21. ↑ E. Landau: Comment on my essay: About the basics of the theory of the faculty series. Meeting reports of the Bavarian Academy of Sciences 39, p. 7
22. ↑ Nörlund, op.cit., P. 258, p. 262 ff
23. ↑ Milne-Thomson, op.cit., P. 287, p. 297
24. ↑ Milne-Thomson, op. Cit., Pp. 287-288