Chebychev interpolations of the Gamma andPolygamma Functions and their analytical

propertiesKarl Dieter Reinartz

May 11, 2016

in memoriamCornelius Lanczos[5] 1893-1974 1

address:

Email: KD.Reinartz@T-Online.de

Kieferndorfer Weg 30, D-91315 Höchstadt, GERMANY

keywords:Gamma function, Sterling formula, Bernoulli numbers, Chebychev approximations, Chebyshevpolynomials, Shifted Chebyshev polynomials, Invers Gamma function, Psi (Digamma) func-tion, Harmonic function, Polygamma functions, Summation/Differentiation/Multiplication ofChebyshev approximations.

Contents1 Introduction 2

2 Chebyshev Interpolations of the Γ Function 32.1 The Γ−1 Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 The LnΓ Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 The Chebyshev Interpolation of the Psi (Digamma) Function 73.1 Summation of the Harmonic Series . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4 Interpolating further Polygamma Functions 74.1 Summation of the higher Harmonic Series . . . . . . . . . . . . . . . . . . . . . . 8

5 Relations of the Shifted Chebychev Polynomials 85.1 Chebychev approximation of smooth functions . . . . . . . . . . . . . . . . . . . 8

5.1.1 Numerical determination of the a∗r-coefficients . . . . . . . . . . . . . . . . 91 http://www.youtube.com/watch?v=avSHHi9QCjAhttp://www.youtube.com/watch?v=PO6xtSxB5Vg

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5.1.2 Summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95.1.3 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95.1.4 Multiplication of two Chebyshev approximations . . . . . . . . . . . . . . 9

1 IntroductionThe Gamma Function derived by Leonhard Euler (1729) is the generalization of discrete facto-rials:

z! = Γ (z + 1) =∫ ∞

0tz−1e−tdt, <z > 0 (1)

The numerical evaluation is not easy. Whittaker+Watson [9] and Temme [8] give a good dis-cussion of the Γ Function and several basic properties.In contrast to that Lanczos [4] developed approximations with a restricted precision by usingChebyshev polynomials in the range [-1..+1] (instead of shifted polynomials which will be usedexclusively in this paper). Chebychev polynomials were introduced into numerical analysis es-pecially by Lanczos [5]in the US since 1935 and by Clenshaw [2] in GB since 1960.–

The next formula is due to James Stirling (1730)

ln Γ (z) ∼ ln(√

2π zz− 12 e−z

)+∞∑

n=1

B2 n

2n (2n− 1) z2 n−1 (2)

The B2 n are the Bernoulli numbers with a poor behaviour:

16 , −

130 ,

142 , −

130 ,

566 ,

6912730 ,

76 , −

3617510 ,

43867798 , −174611

330 ,854513

138 ,−2363640912730 ,

85531036 , −23749461029

870 ,8615841276005

14322 ,7709321041217

510 ,2577687858367

6 , ...(3)

They decrease at the beginning only slowly and then grow with (2n)!. The complete terms inthe infinite sum eq. (2) depend on n and z and grow nevertheless, especially if z is small:

112 · z , −

1360 · z3 ,

11260 · z5 , −

11680 · z7 ,

11188 · z9 , −

691360360 · z11 ,

1156 · z13 , −

3617122400 · z15 ,

43867244188 · z17 ,−

174611125400 · z19 ,

776835796 · z21 , −

2363640911506960 · z23 ,

657931300 · z25 , −

339278014793960 · z27 ,

17231682552012492028 · z29 , −7709321041217

505920 · z31 ,151628697551

396 · z33 , −263152715530534773732418179400 · z35 , ...

(4)

The summation has to stop before the terms begin to grow unrestricted. There is anoptimal position depending on z where summation has to end. This problem is discussed insome detail in[3]2 .A further disadvantage is the low convergence of the admitted terms. In fig. 1 the problemis described in some detail depending on z: fig. 1a shows the maximal number of convergentterms, fig. 1b shows the maximal achievable accuracy in decimal digits.

2...page 467

2

(a) summation limit (b) correct decimal digits

Figure 1: restricted summation and limited precision

2 Chebyshev Interpolations of the Γ FunctionThe Chebyshev polynomials were derived by the Russian Mathematician P. L. Chebyshev (1821-1894) [6]. Among all normalized power polynomials of same degree they have the smallestdeviation from zero in a predefined intervall. Most of their beautiful properties are describedby Snyder [7] and Clenshaw [2] showing many applications to transcendental functions anddifferential equations.The approximation of the Γ Function is represented by

(z − 1)! ' Γ (z) =√

2π zz− 12 e−z ∗

∞0

∑′

a∗rT∗r(1z

) , 1 ≤ z ≤ ∞ (5)

Figure 2 contains 53 Chebyshev coefficients of the Γ-function for an accuracy of 30 decimaldigits in the whole range 1 ≤ z ≤ ∞. Using two coefficients the corresponding powerseries is:

Γ (z) =√

2π · zz− 12 · e−z ∗ [0.999935 + 0.0845506

z] (6)

the maximal relativ error(Figure 3a) is less than 8 ∗ 10−4. U̇sing eleven coefficients for thecorresponding powerseries

Γ (z) =√

2π · zz− 12 · e−z

∗ [0.99999999998 + 0.083333337647z

+ 0.0034720552506z2 − 0.0026788696285

z3

+ 0.00024711193390z4 + 0.00084986066787

z5 − 0.000035855790507z6 − 0.00068599470338

z7

+ 0.00067284352663z8 − 0.00029536102066

z9 + 0.000052647439438z10 ]

(7)

the maximal relativ error(Figure 3b) is less than 2 ∗ 10−11.

3

Chebychev coefficients Γ-function 30 digits

r��∗

r��∗

0+

2.084415923538053

986642658837738

27

-0.00000

00000000000000258766

77518

1+

0.042275268921800

684221304767803

28

+13204101565

2+

0.000005905456639

988647088023171

29

-44391

12769

3-

0.000056497110599

516283178411341

30

+9809

16518

4+

498744212957611

89474

09608

31

-341

34737

5-

103374677775244721

80884

32

-1009

72407

6-

781102997549159904

98736

33

+647

50292

7+

144985544478368664

72503

34

-262

83259

8-

9488746041409554

86866

35

+76

43533

9-

2450853231349408

76160

36

-12

31005

10

+1004170216329377

96250

37

-271738

11

-183302677934966

77815

38

+346373

12

+8672187799785

76352

39

-185684

13

+6796834696249

25584

40

+71428

14

-2882507538880

29128

41

-19923

15

+644507902393

11432

42

+2625

16

-58446607773

01642

43

+1262

17

-21917993063

35859

44

-1269

18

+13605116999

80453

45

+677

19

-42079

61997

50644

46

-269

20

+7708

93543

63348

47

+80

21

+633052293838

48

-12

22

-728

17978

66527

49

-4

23

+352

29384

07205

50

+5

24

-106

54216

31145

51

-3

25

+193634717582

52

+1

26

+92269

52412

Figure 2: Chebyshev coefficients Γ-function 30 digits

2.1 The Γ−1 FunctionFigure 4 contains 53 Chebyshev coefficients of the Γ−1-function for an accuracy of 30 decimaldigits in the whole range 1 ≤ z ≤ ∞.

1(z − 1)! ' Γ−1 (z) = 1√

2πz

12−z ez ∗

∞0

∑′

b∗rT∗r(1z

) , 1 ≤ z ≤ ∞ (8)

4

(a) relative error using only 2 coefficients (b) relative error using 11 coefficients

Figure 3: overall convergence

With four coefficients the powerseries expansion is:

Γ−1 (z) = 1√2π

z12−z ez ∗ [1.000006− 0.08354413

z+ 0.004512425

z2 + 0.001168239z3 ] (9)

The maximal relativ error (Figure 5) is less than 7 ∗ 10−6.In contrast to that in the famous Handbook of Mathematical Functions [1] 3 the series expansionfor Γ−1 is completely wrong.

2.2 The LnΓ FunctionThe approximation is represented by

ln[(z − 1)!] ' ln Γ (z) = ln(√

2π zz− 12 e−z

)+∞0

∑′

c∗rT∗r(1z

) , 1 ≤ z ≤ ∞ (10)

Figure 6 contains the Chebyshev coefficients with a precision of 30 decimal digits for the wholerange of 1 ≤ z ≤ ∞. Using only the first two coefficients and building the power series form

ln Γ (z) = ln√

2π +(z − 1

2

)ln (z)− z + 0.91932 + 0.081160

z(11)

the maximum absolute error (Figure 7a) is less than 5 ∗ 10−4. Using five coefficients

ln Γ (z) = ln√

2π +(z − 1

2

)ln z − z

+ 0.918935 + 0.0833326z

+ 0.000037082z2 − 0.00305155

z3 + 0.000743418z4

(12)

the maximum absolute error (Figure 7b) is less than 2 ∗ 10−7.

3...page 256

5

Chebyshev

coefficientsࢣି-function,

30

digits

r∗

r∗

0+

1.92058

34762

54948

20024

18291

34328

27

+0.0000000000

0000000002

4557198296

1-

0.03896

82376

24892

01214

15331

82954

28

-1

3344512573

2+

0.00078

30980

24559

58872

29636

11440

29

+4658360719

3+

0.00003

65074

79093

94718

63219

85541

30

-1086926183

4-

63995

52363

25099

03462

62930

31

+68916332

5+

2508

67644

41246

12497

57714

32

+93445505

6+

664

29088

71454

74619

02130

33

-64545386

7-

164

72719

66800

61015

61714

34

+27090508

8+

15

22654

0329235267

51875

35

-8155332

9+

178294

7678020568

75297

36

+1439319

10

-1

02239

7351375719

08376

37

+2

10256

11

+21131

54880

28419

64665

38

-3

35833

12

-1579

76630

19572

63953

39

+1

87421

13

-589

22900

53182

13064

40

-74017

14

+294

26575

02392

08743

41

+21359

15

-71

44837

50686

73310

42

-3194

16

+8

04021

40306

78725

43

-1095

17

+1

79571

80338

19450

44

+1242

18

-1

35094

39033

25037

45

-684

19

+44531

0400416213

46

+278

20

-8839

2150224802

47

-85

21

+244

9200580535

48

+15

22

+681

0693615141

49

+4

23

-356

9702427690

50

-5

24

+113

1165397918

51

+3

25

-22

2017658614

52

-1

26

-11809

99223

Figure 4: Chebyshev coefficients Γ−1-function 30 digits

6

Figure 5: Γ−1-function error analysis

3 The Chebyshev Interpolation of the Psi (Digamma) FunctionThis function is the first derivative of the LnΓ Function:

ψ(0)(z) = ψ(z) = ln′ Γ (z) = Γ′(z)Γ(z) = ln z − 1

2z +∞0

∑′

c∗rT∗′r (1z

) , 1 ≤ z ≤ ∞ (13)

After differentiating the sum using eq. (23) and eq. (24) , − 12z = −1

4 ∗ (T ∗0 (1z ) + T ∗1 (1

z )) has tobe added. The final result is

ψ(0)(z) = ψ(z) = Γ′(z)Γ(z) = ln z +

∞0

∑′

α∗(0)rT∗r(1z

) , 1 ≤ z ≤ ∞ (14)

3.1 Summation of the Harmonic Series−ψ(0)(1) = γ = 0.57721 56649 01532 86061 is Euler’s constant.

ψ(0)(n+ 1) − ψ(0)(1) = Hn = 1 + 12 + 1

3 + ...+ 1n , n ∈ N

defines and computes the n-th harmonic number Hn.

4 Interpolating further Polygamma FunctionsDifferentiating the result of eq. (14) as before one gets

ψ(1)(z) = ψ′(z) = 1

z+∞0

∑′

α∗(0)rT∗′r (1z

) , 1 ≤ z ≤ ∞ (15)

7

Finally 1z = 1

2 ∗ (T ∗0 (1z ) + T ∗1 (1

z )) has to be added yielding

ψ(1)(z) =∞0

∑′

α∗(1)rT∗r(1z

) , 1 ≤ z ≤ ∞ (16)

The higher Polygamma Functions can be approximated applying the two step differentiation re-peatedly without additional correction. Each next generated function looses about two decimaldigits in precision.

4.1 Summation of the higher Harmonic SeriesThe general relation is:

(−1)m+1

m! ψ(m)(z) =∑∞

k= 01

(z + k)m+1 = 1zm+1 + 1

(z + 1)m+1 + 1(z + 2)m+1 + ... (17)

and especially for z=n integer

(−1)m+1

m! [ψ(m)(1)− ψ(m)(n)] = 11m+1 + 1

2m+1 + 13m+1 + ... + 1

(n− 1)m+1 (18)

and further specialized with m=1

ψ(1)(1)− ψ(1)(n) = 112 + 1

22 + 132 + ... + 1

(n− 1)2 (19)

5 Relations of the Shifted Chebychev PolynomialsThe Shifted Chebyshev polynomials are defined by

T ∗r (θ) = cos(rθ) − 1 ≤ cos θ ≤ +1 −1 ≤ T ∗r ≤ +12z − 1 = cos θ 0 ≤ z ≤ +1

They are power polynomials in z. Their highest coefficient 2r−1 is used for normalization.Chebyshev proved [6] that among all normalized power polynomials of same degree (or less)they have the smallest deviation from zero in the range 0 ≤ z ≤ +1. That makes them uniquefor optimal interpolation in the declared region. The ranges may be adapted by linear or evennonlinear transformations.The polynomials for the intervall 0 ≤ z ≤ +1 are called the shifted polynomials. They areused here exclusively. Explicit expressions for the first few shifted Chebyshev polynomials are:T ∗0 (z) = 1, T ∗1 (z) = 2z − 1, T ∗2 (z) = 8z2 − 8z + 1, T ∗3 (z) = 32z3 − 48z2 + 18z − 1, ...

Inversion gives:1 = T ∗0 (z), 2z = T ∗0 (z) + T ∗1 (z), 8z2 = 3T ∗0 (z) + 4T ∗1 (z) + T ∗2 (z),32z3 = 10T ∗0 (z) + 15T ∗1 (z) + 6T ∗2 (z) + T ∗3 (z),...

5.1 Chebychev approximation of smooth functions

f(z) =n

0

∑′

a∗rT∗r(z) = 1

2a∗0 + a∗1T

∗1(z) + a∗2T

∗2(z) + a∗3T

∗3(z) + ... (20)

8

5.1.1 Numerical determination of the a∗r-coefficients

For the given function f(z) the a∗r can be determined by

a∗r =m

j=0

∑′′

f(cos2( jπ2m)) cos(rjπm

) (21)

′′ means: terms with j=0 and j=m must be halfed and m should be chosen sufficiently largefor a good approximation.

5.1.2 Summation

1. substituting the T ∗-polynomials by their powerseries representations and thereafter ap-plying the Horner Scheme or

2. it is better to use the coefficients directly: starting with a sufficiently large index n andapplying recursion:

b∗r = (2 ∗ z − 1) ∗ b∗r+1 − b∗r+2 + a∗r , b∗n+1 = b∗n+2 = 0 , r = n, n− 1, ..., 0 (22)

f(z) = 12(b∗0 − b∗2)

5.1.3 Differentiation

1. In order to get f ′(z) =n−10

∑′

a∗′

r T∗r(z) from eq. (20) one starts with a sufficiently large

index r=na∗

′r−1 = a∗

′r+1 + 4 r a∗r , a∗

′n = 0, a∗

′n+1 = 0 (23)

and applies the recursion till r=1.

2. Differentiation (chainrule) of

f(x) =n

0

∑′

a∗rT∗r(x)with x = 1

z results in

f′(z) = − 1

z2 ∗n−10

∑′

a∗′

r T∗r(1z

)

In addition to the former derivation step each coefficient of the derived form has to bemultiplied by

− 1z2 = −(3

8T∗0 (1z

) + 12T∗1 (1z

) + 18T∗2 (1z

)) (24)

applying the multiplication rule of the next subsection.

5.1.4 Multiplication of two Chebyshev approximations

The relation

T ∗m(z) ∗ T ∗n(z) = 12 ∗ [T ∗m+n(z) + T ∗|m−n|(z)] (25)

9

is used for multiplying two polynomials eq. (20). The resulting polynomial has m+n+1 coef-ficients and may be further reduced in length with a minor loss in accuracy.

References[1] Abramowitz, Milton (Hrsg.) ; Stegun, Irene A. (Hrsg.): Handbook of Mathematical Func-

tions with Formulas, Graphs, and Mathematical Tables. U.S. Government Printing Office,Washington, D.C., 1964 (National Bureau of Standards Applied Mathematics Series 55). –xiv+1046 S. – Corrections appeared in later printings up to the 10th Printing, December,1972. Reproductions by other publishers, in whole or in part, have been available since 1965.

[2] Clenshaw, C. W.: Chebyshev Series for Mathematical Functions. London : Her Majesty’sStationery Office, 1962 (National Physical Laboratory Mathematical Tables, Vol. 5. Depart-ment of Scientific and Industrial Research). – iv+36 S.

[3] Graham, Ronald L. ; Knuth, Donald E. ; Patashnik, Oren: Concrete Mathematics:A Foundation for Computer Science. 2nd. Reading, MA : Addison-Wesley PublishingCompany, 1994. – xiv+657 S. – ISBN 0–201–55802–5

[4] Lanczos, Cornelius: A Precision Approximation of the Gamma Function. (1964)

[5] Lanczos, Cornelius ; Bennett, Dr. Albert A. (Hrsg.): Applied Analysis. Prentice Hall,Inc., 1972 (PRENTICE-HALL MATHEMATICS SERIES)

[6] Natanson, Isidor P.: Konstruktive Funktionentheorie. Akademie-Verlag Berlin, 1955

[7] Snyder, Martin A.: Chebyshev Methods in Numerical Approximation. PRENTICE-HALL,INC., 1966 (Series in Automatic Computation)

[8] Temme, Nico M.: Special Functions: An Introduction to the Classical Functions of Math-ematical Physics. New York : John Wiley & Sons Inc., 1996. – xiv+374 S. – ISBN0–471–11313–1

[9] Whittaker, E. T. ; Watson, G. N.: A Course of Modern Analysis. 4th. CambridgeUniversity Press, 1927. – Reprinted in 1996. Table errata: Math. Comp. v. 36 (1981), no.153, p. 319.

10

Chebyshev coefficients LnΓ-function 30 digits

r��∗

r��∗

0+

0.081859815904678

13286

7937908794

27

-0.00000

00000000000000252444

70512

1+

0.040579741747366

64225

7034689356

28

+13281492164

2-

0.000404907931144

69879

9131629808

29

-4549724058

3-

0.000048897289375

19193

1988707281

30

+1033696592

4+

58079

53583619250489236122

31

-51294655

5-

1263

98282718429693139424

32

-97313164

6-

734

49134703220165685808

33

+64680725

7+

155

71158549445243897676

34

-26693664

8-

123059452195

28798

08979

35

+7899455

9-

21416116407

73544

01689

36

-1334338

10

+10169697689

38973

88713

37

-241499

11

-19745

33328

7229064398

38

+341305

12

+1215

14210

5272646241

39

-186612

13

+637

30023

8999434772

40

+72733

14

-291

75788

3116352078

41

-20640

15

+67

976604799256568

42

+2907

16

-69258621512

51366

43

+1180

17

-20012646187

92255

44

-1256

18

+13575674519

07062

45

+681

19

-4332938804

96534

46

-274

20

+827125257

24804

47

+82

21

-878351279129

48

-14

22

-705

7211991064

49

-4

23

+354

8912249033

50

+5

24

-109

8579810742

51

-3

25

+207734840963

52

+1

26

+5275541308

Figure 6: Chebyshev coefficients lnΓ-function 30 digits

11

(a) absolute error using only 2 coefficients (b) absolute error using 5 coefficients

Figure 7: overall convergence

Chebyshev coefficients � (� )function 20 digitsr � (� ) �

∗ r � (� ) �∗

0 + 0.44099 88884 92856 94691 15 + 0.00000 00000 00227 981951 + 0.21108 63054 00967 91864 16 - 120 049562 - 0.00912 29231 52255 54182 17 + 32 345303 + 0.00031 04394 40358 50795 18 - 4 605264 + 0.00001 52324 36589 43642 19 - 544355 - 46008 36953 34872 20 + 626346 + 4298 51491 37346 21 - 243777 + 206 19925 15161 22 + 59928 - 156 70624 26952 23 - 6509 + 30 03990 64500 24 - 25810 - 1 94358 59626 25 + 19911 - 72747 46204 26 - 7812 + 31603 48530 27 + 2013 - 6567 37755 28 - 314 + 483 18059 29 - 1

Figure 8: The ψ(0)-approximation 20 digits

12

Chebyshev coefficients � (� )-function 20 digitsr � (� )�

∗ r � (� )�∗

0 - 0.65708 14986 14027 69733 16 - 0.00000 00000 00272 965211 - 0.43109 98074 89671 50892 17 + 78 981692 - 0.09873 66335 87060 00171 18 + 53 570733 + 0.00356 42792 12194 74453 19 - 17 100434 - 0.00025 87473 59549 76014 20 + 3 264925 + 31711 69149 43256 21 - 34796 + 31208 83744 00715 22 + 278557 - 6189 17377 99128 23 + 140098 + 468 49699 44897 24 - 43379 + 87 01863 21026 25 + 82010 - 40 20193 46760 26 - 2111 + 7 75739 90580 27 - 10012 - 47235 75596 28 + 5213 - 25200 70786 29 - 1814 + 11506 14546 30 + 415 - 2679 83664

Figure 9: The ψ(1)-function 20 digits

13