November 2, 2023

Interpolation problems
Michel Waldschmidt
Professeur Émérite, Sorbonne Université,Institut de Mathématiques de Jussieu, Paris
http://www.imj-prg.fr/~michel.waldschmidt/
1 / 31ICSFA 2023 — Abstract
Given a sequence (sn )n≥0 of complex numbers, a sequence(tn )n≥0 of nonnegative integers and a sequence (E n )n≥0 of
subsets of C, the corresponding interpolation problem is toinvestigate the set of entire functions f such thatf (tn ) (sn ) ∈ E n for all n. The existence problem is when eachE n consists of a single element. The unicity problem is for
E n = {0} for all n. The arithmetic problem of integer valued
functions is for E n = Z for all n. Classical examples are thoseinvestigated by Abel (1802–1829, published posthumously in1881), Halphén (1882), Gontcharoff (1930), Lidstone (1932),Poritsky (1932), Whittaker (1934), Schoenberg (1936), Straus(1950), Macintyre (1954), Buck (1955), Sato (1964). Many
problems are open; we discuss some of them.

2 / 31ICSFA 2016
2016, September 9-11, XVth Annual Conference of Society forSpecial Functions and their Applications (ICSFA-2016),
Department of Applied Sciences and Humanities, Faculty ofEngineering and Technology, Jamia Millia Islamia (CentralUniversity), New Delhi.
September 9, 2016 — R. P. Agarwal Memorial Lecture:Irrationality and transcendence of values of special functions.
Proceedings, New Delhi, India. Editors: A. K. Agarwal, M. A.
Pathan, Subuhi Khan, Ch. Wali Mohd.
http://www.ssfaindia.org/conf_proc.htm
http://www.imj-prg.fr/~michel.waldschmidt/AgendaArchives.html

3 / 31ICSFA 20182018, November 22 - 24: 17th Annual Meeting of SSFA International Conference on Special Functions & Applications(ICSFA-2018), Amal Jyothi College on Engineering,Kanjirapalli, Kottayam (Kerala).
Lecture on Linear recurrence sequences: an introduction.
November 23, 2018: M. A. Pathan, AMU, Aligarh.
On Generalization of Taylor’s series, Riemann Zeta Functions
and Bernoulli Polynomials

http://www.imj-prg.fr/~michel.waldschmidt/AgendaArchives.html

4 / 31ICSFA 2019
October 21 - 23, 2019: Bikaner, RajasthanInternational Conference on Special Functions & Applications(ICSFA-2019). Department of Mathematics, University
College of Engineering and Technology.
Lecture on Lidstone series, generalisations and arithmetic
applications.

http://www.imj-prg.fr/~michel.waldschmidt/AgendaArchives.html

5 / 31ICSFA 2021 (online)
Decembre 22-23, 2021, Department of Mathematics,
University of Kerala
ICSFA-2021 (International Conference on Special Functionsand Applications) and 20th Annual meeting of the Society forSpecial functions and their Applications.
Lecture on Lidstone expansion in several variables.

http://www.imj-prg.fr/~michel.waldschmidt/AgendaArchives.html

6 / 31ICSFA 2022

November 26 - 28, 2022, Department of Studies inMathematics, University of Mysore, India.
International Conference on Special Functions & Applications(ICSFA-2022) (XXIst Annual Meeting of the Society forSpecial Functions & their Applications, India).
Lecture on A multidimensional analog of Lidstoneinterpolation theory. .
http://ssfaindia.org/icsfa2022/

7 / 31ICSFA 2023

November 2, 2023

International Conference on Special Functions & Applications (ICSFA-2023).
http://ssfaindia.org.in/icsfa2023/
Department of Mathematics, Ramanujan School of Mathematical Sciences
Pondicherry University, Puducherry, India

Interpolation Problems
http://www.imj-prg.fr/~michel.waldschmidt/

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Examples of interpolation problems
I(Lidstone):f (2n) (0) = an ,If (2n) (1) = bn

for n ≥ 0.

(Poritsky): For m ≥ 2 and σ0 , . . . , σm−1 in C,f (mn) (σj ) = anj

Ifor n ≥ 0.

(Whittaker):f (2n+1) (0) = an ,If (2n) (1) = bn

for n ≥ 0 and j = 0, 1, . . . , m − 1.

(Gontcharoff): For (σn )n≥0 a sequence of complex
numbers,f (n) (σn ) = an for n ≥ 0.
9 / 31Taylor expansion: vertical
f (n)
f0
f..
.
•..
.
••
0znn!

Hurwitz functions
Kakeya, S. (1916).
Notes on the maximum modulus of a function.
Tohoku Math. J., 10:68–72.
Pólya, G. (1921).
Über die kleinsten ganzen Funktionen, deren sämtliche Derivierten
im Punkte z = 0 ganzzahlig sind. (Auszug aus einem an T. Kubota
gerichteten Briefe).
Tohoku Math. J., 19:65–68.

10 / 31Pólya’s expansion: horizontalf • • • ···0 1 2 ···

1, z,

• ···
m ···

z(z − 1) · · · (z − m + 1)z(z − 1),...,,...
2m!

Integer–valued entire functions. Calculus of finite differences.
Pólya, G. (1915).
Über ganzwertige ganzen Funktionen.
Rend. circ. math. Palermo, 40:1–16.

G.H. Hardy (1917), A. Selberg (1941), C. Pisot (1942).
11 / 312–point Hurwitz functions
..
.
f (n)
..
.
f0f..
.
•..
.

..
.
•..
.

• •
• •
0 1

Gel’fond, A. O. (1934).
Über Potenzreihen mit ganzen Koeffizienten.
Wiss. Ber. Moskauer Univ. 2, 29-33 (1934).

12 / 312–times integer–valued functions
f0 • • • · · ·f • • • ···0 1 2 ···

• ···
• ···
m ···

Gel’fond, A. O. (1929).
Sur les propriétés arithmétiques des fonctions entières.
Tohoku Math. J., 30:280–285.
Selberg, A. (1941).
Über einen Satz von A. Gel’fond.
Arch. Math. Naturvid., 44:159–170.

13 / 31

Utterly integer–valued entire functions
..
.
f (n)
..
.
f0f..
.
•
..
.

..
.
• ···
.. . .
.
.

..
.
•
..
.

···

• • ··· • ···• • ··· • ···0 1 ··· m ···Straus, E. G. (1951).
On the polynomials whose derivatives have integral values at theintegers.
Proc. Amer. Math. Soc., 2:23–27.
Fridman, G. A. (1968).
Entire integer-valued functions.
Mat. Sb. (N.S.), 75 (117):417–431.
14 / 31Abel interpolation
..
.

..

f (n)
..
.
0ff.

•.

..

••
0 1 ···n ···Abel, N. H. (1881).
Œuvres complètes de Niels Henrik Abel. Vol. 1 and 2.
Cambridge: Cambridge University Press, reprint of the new edition
published 1881 by grondahl and son, 2012 edition.
Halphén, G. (1882).
Sur une série d’Abel.
C. R. Acad. Sci., Paris, 93:1003–1005.
Bull. Soc. Math. Fr., 10:67–87.
15 / 31

Lidstone interpolation..
.
f (2n+1)f (2n)
..
.
f 00
f0f..
.
◦
•
..
.

..
.
◦•
..
.

• •◦ ◦• •s0 s1

Lidstone, G. J. (1930).
Notes on the extension of Aitken’s theorem (for polynomialinterpolation) to the Everett types.
Proc. Edinb. Math. Soc., II. Ser., 2:16–19.
16 / 31

Whittaker interpolation..
.
f (2n+1)f (2n)
..
.
f 00
f0f..
.
•
◦
..
.

..
.
◦•
..
.

◦ •• ◦◦ •s0 s1

Whittaker, J. M. (1933).
On Lidstone’s series and two-point expansions of analytic functions.
Proc. Lond. Math. Soc. (2), 36:451–469.
Whittaker, J. M. (1935).
Interpolatory function theory, volume 33.
Cambridge University Press, Cambridge.

17 / 31

Poritsky interpolation (3 points)..
.
f (3n+2)
f (3n+1)
f (3n)..
.
f (iv)
f 000
f 00
f0
f..
.
◦◦
•
..
.

..
.
◦◦
•
..
.

..
.
◦◦
•
..
.

◦ ◦ ◦
• • •
◦ ◦ ◦◦ ◦ ◦• • •s0 s1 s2

Poritsky, H. (1932).
On certain polynomial and other approximations to analyticfunctions.
Trans. Amer. Math. Soc., 34(2):274–331.
18 / 31Gontcharoff interpolation (3 points)..
.
f (3n+2)
f (3n+1)
f (3n)..
.
f (iv)
f 000
f 00
f0
f..
.
◦◦
•
..
.

..
.
◦•
◦
..
.

..
.
•◦
◦
..
.

◦ • ◦
• ◦ ◦◦ ◦ •◦ • ◦• ◦ ◦s0 s1 s2

Gontcharoff, W. (1930).
Recherches sur les dérivées successives des fonctions analytiques.
Généralisation de la série d’Abel.
Ann. Sci. Éc. Norm. Supér. (3), 47:1–78.
19 / 31Abel–Gontcharoff interpolation: f (n)(wn)
We start with any sequence w = (wn )n≥0 of complex numbers.
We define a sequence of polynomials (Ωw0 ,w1 ,...,wn−1 )n≥0 in
C[z] so that any polynomial P can be written as a finite sum
XP (z) =P (n) (wn )Ωn;w (z).
n≥0The condition isΩ(k)n;w (w k ) = δkn
for n ≥ 0 and k ≥ 0.
Alternatively, one may proceed as follows: we set Ω∅ = 1,
Ωw0 (z) = z − w0 , and, for n ≥ 1, we define Ωw0 ,w1 ,w2 ,...,wn (z)as the polynomial of degree n + 1 which is the primitive ofΩw1 ,w2 ,...,wn vanishing at w0 .
20 / 31The sequence of polynomials Ωn;wFor n ≥ 0, we write Ωn;w for Ωw0 ,w1 ,...,wn−1 , a polynomial ofdegree n which depends only on the first n terms of thesequence w. The leading term of Ωn;w is (1/n!)z n .
In particular, for N ≥ 0 we haveNXzN
1=wN −n Ωn;w (z).
N ! n=0 (N − n)! nThis gives an inductive formula defining ΩN ;w : for N ≥ 0,N −1zN X1ΩN ;w (z) =−wN −n Ωn;w (z).
N ! n=0 (N − n)! n

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Iterated integrals
We also have
Ωw0 ,w1 ,...,wn (z) = Ω0,w1 −w0 ,w2 −w0 ,...,wn −w0 (z − w0 ).
With w0 = 0, the first polynomials are given by
2!Ω0,w1 (z) = (z − w1 )2 − w1 2 ,3!Ω0,w1 ,w2 (z) = (z − w2 )3 − 3(w1 − w2 )2 z + w2 3 ,4!Ω0,w1 ,w2 ,w3 (z) = (z − w3 )4 − 6(w2 − w3 )2 (z − w1 )2− 4(w1 − w3 )3 z + 6w1 2 (w2 − w3 )2 − w3 4 .
From the definition we deduce the following formula, involving
iterated integrals
Z z
Z t1
Z tn−1Ωw0 ,w1 ,...,wn−1 (z) =dt1
dt2 · · ·
dtn .
w0

w1

wn−1
22 / 31

A determinant
Gontcharoff:
z1!
w01
1!
1

z2···
2!
w0 2···
2!
w1···
1!

Ωw0 ,w1 ,...,wn−1 (z) = (−1)n 0

1001···

..
.
0..
.
0..
.
0..

.

···

z n−1(n − 1)!
w0 n−1(n − 1)!
w1 n−2(n − 2)!
w2 n−3(n − 3)!
..
.
1znn!
w0 nn!
w1 n−1(n − 1)! .
w2 n−2(n − 2)!
..
.
wn−1
1!

23 / 31

ExamplesWith the sequence w = (1, 0, 1, 0, . . . , 0, 1, . . . ), we recover
the Whittaker polynomials associated with the interpolation
problemf (2n+1) (0) = an ,f (2n) (1) = bn

for n ≥ 0.

Another example, considered by N. Abel is the arithmetic
progression w = (a + nt)n≥0 with a in C and t in C \ {0},where1Ωn;w (z) = (z − a)(z − a − nt)n−1n!
for n ≥ 1, which satisfiesΩ0n;w (z) = Ωn−1;w (z − t).
24 / 31

Standard pairs of polynomials (Whittaker)
Let S ⊂ {0, 1} × N. Let p = (p1 , p2 , . . . ) be the sequence ofintegers such that (1, pi ) ∈ E and let q = (q 1 , q 2 , . . . ) be thesequence of integers such that (0, q i ) ∈ E.
According to Whittaker, a set of polynomials(π, ζ) = {π1 , π2 , . . . , πm , . . . ; ζ1 , ζ2 , . . . , ζm , . . . }is a standard set of polynomials for S if any polynomialf ∈ C[z] can be written (in a unique way)Xf (z) =f (pn ) (1)πn (z) + f (qn ) (0)ζn (z)n≥025 / 31

Equivalent definition
Given two sequences p = (pn )n≥0 and q = (q n )n≥0 of
nonnegative integers, a standard set of polynomials for (p, q) isa pair (π, ζ) given by two sequences π = (πn )n≥0 andζ = (ζn )n≥0 of polynomials such that, for n, k ≥ 0,πn(pk ) (1) = δnk ,πn(qk ) (0) = 0

ζn(pk ) (1) = 0,

ζn(qk ) (0) = δnkand

26 / 31

Whittaker classification
Here is Whittaker classification of complete, indeterminate andredundant sequences, involving standard sets of polynomials.
If there exists a unique standard set of polynomials (π, ζ),then (p, q) is called complete, and any polynomial f can be
written in a unique way as a finite sumXXf (z) =f (pn ) (1)πn (z) +f (qn ) (0)ζn (z).
n≥0n≥0If there are several solutions (π, ζ), then (p, q) is calledindeterminate.
If there is no solution (π, ζ), then (p, q) is called redundant.

27 / 31Complete sets
For m ≥ 1, define
P (m) = #{0 ≤ n ≤ m − 1 ; (0, n) ∈ E}
andQ(m) = #{0 ≤ n ≤ m − 1 ; (1, n) ∈ E}.
Then the set E is complete if and only if
P (m) + Q(m) ≥ m
for all m ≥ 1, and there are infinitely many m withP (m) + Q(m) = m.

28 / 31Indeterminate sets

If the set E is indeterminate, then P (m) + Q(m) < m for
infinitely many m.

29 / 31

Some research problems• In the complete case, study the unique standard set ofpolynomials (π, ζ).
• Extend to more than two points.
• Generalize to several variables.
• Investigate the arithmetic question of integer valued entire
functions in these situations.

30 / 31November 2, 2023

Interpolation problems
Michel Waldschmidt
Professeur Émérite, Sorbonne Université,Institut de Mathématiques de Jussieu, Paris
http://www.imj-prg.fr/~michel.waldschmidt/
31 / 31