Collection of Infinite Products and Series

   Dr. Andreas Dieckmann, Physikalisches Institut der Uni Bonn

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My interest in infinite products started in the year 2000 with the problem of the electrical field of a line charge
trapped inside a rectangular tube. After I learned that the double product can be solved using elliptic theta functions,
my collectors passion was triggered. The site has been growing ever since, and its focus has been expanded
to include series as well.

These pages list thousands of expressions like products, sums, relations and limits shown in the following sections:

-  Infinite Products

-  Products involving Theta Functions

-  Other formulae and curiosities including sums of hyperbolic and inverse tangent (arctan) functions and q - series

-  q-Series

-  special values of EllipticK and EllipticE

-  Series of Hyperbolic Functions

-  Series of CosIntegral

-  some Limits

-  diverse Series

-  Series of Logarithms

-  Series of Inverse Tangents ( Arcustangent )

-  Series of Bessel Functions

-  Series of Legendre Polynomials

-  Series of Zeta PolyGamma PolyLog and related

-  Series of Beta Functions

-  Series of Gamma Functions

-  Series involving HarmonicNumber

-  Series involving Hypergeometric Functions

-  a few Integrals

-  iterated expressions ( Tetration )

-  some properties of ProductLog LerchPhi and PolyLog

{j, n, m} are Integer; {λ, q} > 0 and r are real; {z, InfProd_3.gif, InfProd_4.gif, InfProd_5.gif, InfProd_6.gif} may be complex; Γ[a] is  Gamma[a];
some of the products possess pointlike poles, where the denominator of a factor gets zero for certain
values of z. Some of the expressions are well known, others may be not; some were found in the
depths of the world wide web, the first are derived from the following product below.
- any formula you decide to use should be numerically tested for validity in the users domain -

Infinite Products : ( Back to Top )

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This product converges and delivers infinite product representations for many functions if the {a, b, c, d} are
replaced by constants and simple functions in z :

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Products of two Gammas:

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partial fraction decompositions :

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Eulers product:

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Products involving Theta Functions    ( Back to Top )

InfProd_116.gif is shorthand for EllipticTheta[n, z, q] and InfProd_117.gif means EllipticThetaPrime[m, z, q].

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( The above product was calculated together with D. Zagier, MPI für Mathematik, Bonn. It numerically converges best if k ≫ n. )

Series and product representations :

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With  InfProd_124.gif[ 0 , q ] a few relations between the theta functions are

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These limits give "needles" of height ±1 situated at the extreme values of Cos or Sin respectively (n∼1/(4λ)):

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Partial differential equation :

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EllipticThetas with imaginary argument :

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With z → 0 we get

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Half Lambda:

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Other relations :

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The following double products numerically converge best if k ≫ n.

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Double product representation of the single theta functions:

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If the product over k is carried out first we get products with Tanh and Coth :

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The theta functions may be expressed through each other:

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and exhibit a kind of double periodicity ({m, n} ∈ Integer) :

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With  m = InverseEllipticNomeQ[Exp[-π λ]] and K[m] = EllipticK[m]:   

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In the following is ( 0 < q < 1 ) and InfProd_201.gif[ 0 , q ] ,   (InfProd_202.gif[ 0 , q ] =InfProd_203.gif[ 0 , -q ] ) :

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m = InverseEllipticNomeQ[q], K[m] = EllipticK[InverseEllipticNomeQ[q]] and E[m] = EllipticE[InverseEllipticNomeQ[q]]:

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InverseEllipticNomeQ m[q], K[m[q]] and E[m[q]] expressed through infinite products or theta functions:

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InfProd_233.gifInfProd_234.gif and InfProd_235.gif can be expressed through m[q] , K[m[q]] and E[m[q]] :

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and similarly :

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and:

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and from combining the above like:

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we get:

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as q is getting larger than InfProd_241.gif the branch cut of K and E is crossed, so the continuous and smooth complex functions are built from two parts :

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If the result of the imaginary transformation doesn't seem right, consider the following points:
• If in the resulting formula a sign change of the imaginary part as function of q occurs under a square root ( at q = Exp[- π / 2] ) then the square root may take the other sign
• Logs with complex arguments may end up on a wrong branch, try replacing Log[...] with Log[...] + n I 2 π

Theta Functions (z = InfProd_243.gif) expressed through EllipticK and m :

InfProd_244.gif InfProd_245.gif InfProd_246.gif
InfProd_247.gif InfProd_248.gif InfProd_249.gif
InfProd_250.gif InfProd_251.gif InfProd_252.gif
InfProd_253.gif InfProd_254.gif InfProd_255.gif
InfProd_256.gif InfProd_257.gif InfProd_258.gif
InfProd_259.gif InfProd_260.gif InfProd_261.gif
InfProd_262.gif InfProd_263.gif InfProd_264.gif
InfProd_265.gif InfProd_266.gif InfProd_267.gif

Series expansion of InverseEllipticNomeQ:

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Series expansion of EllipticNomeQ:

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specific values:

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For the general expression see below near the end of the paragraph.

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Theta Functions , specific values :

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The missing ϑ functions we get from

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Other formulae and curiosities including sums of hyperbolic and inverse tangent (arctan) functions and q - series:   ( Back to Top )

The following double sums numerically converge best if k ≫ n. For numerical checks the finite lower limit should be increased by 1 in case the running index is shifted by -1/2.
ϑ ‘’ represents the second derivative of ϑ (x,q) with respect to x, eg: InfProd_303.gif means InfProd_304.gif.

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The following approximations hold to about 2% over all a :

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q - Series :   ( Back to Top )

with InfProd_474.gif → Cosh[ k Log[ q ]] + Sinh[ k Log[ q ]] the following expressions can be transformed into sums of hyperbolic functions.

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The inner sum above gives the number of ascending sequences of length k in the permutations of n numbers.
For natural n PolyLog[-n, q] appears as a rational function in q.

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There is a small inconsistency in the definition of LerchPhi[q,0,0]: LerchPhi[q, 0, 0] = 1/(1 - q) whereas LerchPhi[q, n, 0] /. n -> 0 = q/(1 - q).

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( m = InverseEllipticNomeQ[q], K[m] = EllipticK[m], E[m] = EllipticE[m] ):

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The appearing of n or n - 1 as summation stop index implies n ∈ Integer.

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The introduction of QPolyGamma[n, z, q] (nth derivative of QDigamma function (z, q)) in Mathematica 7 allows expression of

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qHypergeometric2F1[a, b, c, q, x] is a q-variant of 2F1:

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QPolyGamma and QPochhammer Identities :

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Real Part Imaginary Part
n=1: InfProd_672.gif 0
n=2: InfProd_673.gif InfProd_674.gif
n=3: InfProd_675.gif InfProd_676.gif
n=4: InfProd_677.gif InfProd_678.gif
n=5: InfProd_679.gif InfProd_680.gif
n=6: InfProd_681.gif InfProd_682.gif
n=7: InfProd_683.gif InfProd_684.gif

special values of EllipticK and EllipticE:   ( Back to Top )

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E[m] is EllipticE[m];

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Series of Hyperbolic Functions:   ( Back to Top )

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InfProd_721.gif

InfProd_722.gif

ϑ ‘’ represents the second derivative of ϑ(x,q) with respect to x, eg: InfProd_723.gif means InfProd_724.gif.

InfProd_725.gif

InfProd_726.gif

For numerical checks the finite lower negative limit should be increased by 1 where the running index is shifted by - 1/2.

InfProd_727.gif

InfProd_728.gif

InfProd_729.gif

InfProd_730.gif

InfProd_731.gif

InfProd_732.gif

m = InverseEllipticNomeQ[InfProd_733.gif] :

InfProd_734.gif

InfProd_735.gif

InfProd_736.gif

InfProd_737.gif

InfProd_738.gif

InfProd_739.gif

InfProd_740.gif

InfProd_741.gif

InfProd_742.gif

InfProd_743.gif

InfProd_744.gif

InfProd_745.gif

InfProd_746.gif

InfProd_747.gif

InfProd_748.gif

InfProd_749.gif

InfProd_750.gif

The next expression uses an idea taken from “Weiss, J.D.(2014) The Summation of One Class of Infinite Series. Applied Mathematics, 5, 2815 - 2822. http://dx.doi.org/10.4236/am.2014.517269”:

InfProd_751.gif

InfProd_752.gif

InfProd_753.gif

InfProd_754.gif

InfProd_755.gif

InfProd_756.gif

InfProd_757.gif

InfProd_758.gif

InfProd_759.gif

InfProd_760.gif

InfProd_761.gif

InfProd_762.gif

InfProd_763.gif

InfProd_764.gif

InfProd_765.gif

InfProd_766.gif

InfProd_767.gif

InfProd_768.gif

InfProd_769.gif

InfProd_770.gif

InfProd_771.gif

InfProd_772.gif

InfProd_773.gif

InfProd_774.gif

InfProd_775.gif

InfProd_776.gif

InfProd_777.gif

InfProd_778.gif

InfProd_779.gif

InfProd_780.gif

InfProd_781.gif

InfProd_782.gif

InfProd_783.gif

InfProd_784.gif

InfProd_785.gif

InfProd_786.gif

Series of CosIntegral:   ( Back to Top )

InfProd_787.gif

InfProd_788.gif

InfProd_789.gif

InfProd_790.gif

InfProd_791.gif

InfProd_792.gif

InfProd_793.gif

InfProd_794.gif

InfProd_795.gif

InfProd_796.gif

InfProd_797.gif

InfProd_798.gif

some Limits :   ( Back to Top )

InfProd_799.gif

InfProd_800.gif

InfProd_801.gif

InfProd_802.gif

InfProd_803.gif

diverse Series :   ( Back to Top )

InfProd_804.gif

InfProd_805.gif

InfProd_806.gif

InfProd_807.gif

InfProd_808.gif

InfProd_809.gif

InfProd_810.gif

InfProd_811.gif

InfProd_812.gif

InfProd_813.gif

InfProd_814.gif

InfProd_815.gif

The appearing of n or m as summation stop index implies n,m ∈ Integer.

InfProd_816.gif

InfProd_817.gif

InfProd_818.gif

InfProd_819.gif

InfProd_820.gif

InfProd_821.gif

InfProd_822.gif

InfProd_823.gif

( Zeta[n - 1] - Zeta[n] = Zeta[n - 1, 0] - Zeta[n, 0] = Zeta[n - 1, 1] - Zeta[n, 1] = Zeta[n - 1, 2] - Zeta[n, 2],   Zeta[n,1] = Zeta[n,0] = Zeta[n] )

InfProd_824.gif

InfProd_825.gif

InfProd_826.gif

InfProd_827.gif

InfProd_828.gif

InfProd_829.gif

InfProd_830.gif

InfProd_831.gif

InfProd_832.gif

InfProd_833.gif

InfProd_834.gif

InfProd_835.gif

InfProd_836.gif

InfProd_837.gif

The sum  InfProd_838.gif  gives following results for some rational s :

InfProd_839.gif

InfProd_840.gif

InfProd_841.gif

InfProd_842.gif

InfProd_843.gif

InfProd_844.gif

InfProd_845.gif

InfProd_846.gif

InfProd_847.gif

InfProd_848.gif

InfProd_849.gif

InfProd_850.gif

InfProd_851.gif

InfProd_852.gif

InfProd_853.gif

InfProd_854.gif

InfProd_855.gif

InfProd_856.gif

InfProd_857.gif

InfProd_858.gif

InfProd_859.gif

InfProd_860.gif

InfProd_861.gif

This sum alternates between ± π for z ∈ N :

InfProd_862.gif

In the following 4 expressions b =InfProd_863.gif :

InfProd_864.gif

InfProd_865.gif

InfProd_866.gif

InfProd_867.gif

InfProd_868.gif

InfProd_869.gif

InfProd_870.gif

The next three expressions contain s = InfProd_871.gif and t = InfProd_872.gif:

InfProd_873.gif

InfProd_874.gif

InfProd_875.gif

InfProd_876.gif

InfProd_877.gif

InfProd_878.gif

InfProd_879.gif

InfProd_880.gif

InfProd_881.gif

InfProd_882.gif

InfProd_883.gif

InfProd_884.gif

InfProd_885.gif

InfProd_886.gif

InfProd_887.gif

InfProd_888.gif

InfProd_889.gif

InfProd_890.gif

InfProd_891.gif

InfProd_892.gif

InfProd_893.gif

InfProd_894.gif

InfProd_895.gif

InfProd_896.gif

InfProd_897.gif

Series of Logarithms :   ( Back to Top )

(m = InverseEllipticNomeQ[q], K[m] = EllipticK[m], E[m] = EllipticE[m]), the appearing of n or n - 1 as summation stop index implies n ∈ Integer.

InfProd_898.gif

InfProd_899.gif

InfProd_900.gif

InfProd_901.gif

InfProd_902.gif

InfProd_903.gif

InfProd_904.gif

InfProd_905.gif

InfProd_906.gif

InfProd_907.gif

InfProd_908.gif

InfProd_909.gif

InfProd_910.gif

InfProd_911.gif

InfProd_912.gif

InfProd_913.gif

InfProd_914.gif

InfProd_915.gif

InfProd_916.gif

InfProd_917.gif

InfProd_918.gif

InfProd_919.gif

InfProd_920.gif

InfProd_921.gif

InfProd_922.gif

InfProd_923.gif

InfProd_924.gif

InfProd_925.gif

InfProd_926.gif

InfProd_927.gif

InfProd_928.gif

InfProd_929.gif

InfProd_930.gif

InfProd_931.gif

InfProd_932.gif

InfProd_933.gif

InfProd_934.gif

InfProd_935.gif

InfProd_936.gif

InfProd_937.gif

InfProd_938.gif

InfProd_939.gif

InfProd_940.gif

InfProd_941.gif

InfProd_942.gif

InfProd_943.gif

InfProd_944.gif

InfProd_945.gif

InfProd_946.gif

InfProd_947.gif

InfProd_948.gif

InfProd_949.gif

InfProd_950.gif

InfProd_951.gif

InfProd_952.gif

InfProd_953.gif

InfProd_954.gif

InfProd_955.gif

InfProd_956.gif

InfProd_957.gif

InfProd_958.gif

InfProd_959.gif

InfProd_960.gif

InfProd_961.gif

InfProd_962.gif

InfProd_963.gif

InfProd_964.gif

InfProd_965.gif

InfProd_966.gif

InfProd_967.gif

InfProd_968.gif

InfProd_969.gif

InfProd_970.gif

InfProd_971.gif

InfProd_972.gif

InfProd_973.gif

InfProd_974.gif

InfProd_975.gif

InfProd_976.gif

Next is the 'Fountain' function, plot it in the range of -50 < z < 10 with parametervalues of a between -3 and 1 !

InfProd_977.gif

InfProd_978.gif

InfProd_979.gif

InfProd_980.gif

InfProd_981.gif

InfProd_982.gif

InfProd_983.gif

InfProd_984.gif

InfProd_985.gif

InfProd_986.gif

InfProd_987.gif

InfProd_988.gif

InfProd_989.gif

InfProd_990.gif

InfProd_991.gif

InfProd_992.gif

InfProd_993.gif

InfProd_994.gif

InfProd_995.gif

InfProd_996.gif

InfProd_997.gif

InfProd_998.gif

InfProd_999.gif

InfProd_1000.gif

InfProd_1001.gif

InfProd_1002.gif

InfProd_1003.gif

InfProd_1004.gif

Series of Inverse Tangents ( Arcustangent ) :   ( Back to Top )

(m = InverseEllipticNomeQ[q], K[m] = EllipticK[m], E[m] = EllipticE[m]), the appearing of n or n - 1 as summation stop index implies n ∈ Integer.

InfProd_1005.gif

InfProd_1006.gif

InfProd_1007.gif

InfProd_1008.gif

InfProd_1009.gif

InfProd_1010.gif

InfProd_1011.gif

InfProd_1012.gif

InfProd_1013.gif

InfProd_1014.gif

InfProd_1015.gif

InfProd_1016.gif

InfProd_1017.gif

InfProd_1018.gif

InfProd_1019.gif

InfProd_1020.gif

InfProd_1021.gif

InfProd_1022.gif

InfProd_1023.gif

LogGamma[z] is used, because it has a simpler branch strucure than Log[Gamma[z]] and avoids many discontinuities.

InfProd_1024.gif

InfProd_1025.gif

InfProd_1026.gif

InfProd_1027.gif

InfProd_1028.gif

InfProd_1029.gif

InfProd_1030.gif

InfProd_1031.gif

InfProd_1032.gif

InfProd_1033.gif

InfProd_1034.gif

InfProd_1035.gif

InfProd_1036.gif

InfProd_1037.gif

InfProd_1038.gif

InfProd_1039.gif

InfProd_1040.gif

InfProd_1041.gif

InfProd_1042.gif

InfProd_1043.gif

InfProd_1044.gif

InfProd_1045.gif

InfProd_1046.gif

InfProd_1047.gif

InfProd_1048.gif

InfProd_1049.gif

InfProd_1050.gif

InfProd_1051.gif

InfProd_1052.gif

Some ArcTan Identities :

InfProd_1053.gif

Series of Bessel Functions :   ( Back to Top )

InfProd_1054.gif

InfProd_1055.gif

InfProd_1056.gif

InfProd_1057.gif

InfProd_1058.gif

InfProd_1059.gif

InfProd_1060.gif

InfProd_1061.gif

InfProd_1062.gif

InfProd_1063.gif

InfProd_1064.gif

InfProd_1065.gif

InfProd_1066.gif

InfProd_1067.gif

InfProd_1068.gif

InfProd_1069.gif

InfProd_1070.gif

InfProd_1071.gif

InfProd_1072.gif

InfProd_1073.gif

InfProd_1074.gif

InfProd_1075.gif

InfProd_1076.gif

InfProd_1077.gif

InfProd_1078.gif

InfProd_1079.gif

InfProd_1080.gif

InfProd_1081.gif

InfProd_1082.gif

InfProd_1083.gif

InfProd_1084.gif

InfProd_1085.gif

InfProd_1086.gif

InfProd_1087.gif

InfProd_1088.gif

InfProd_1089.gif

InfProd_1090.gif

InfProd_1091.gif

InfProd_1092.gif

InfProd_1093.gif

InfProd_1094.gif

InfProd_1095.gif

InfProd_1096.gif

InfProd_1097.gif

InfProd_1098.gif

InfProd_1099.gif

InfProd_1100.gif

InfProd_1101.gif

InfProd_1102.gif

InfProd_1103.gif

InfProd_1104.gif

InfProd_1105.gif

InfProd_1106.gif

InfProd_1107.gif

InfProd_1108.gif

Series of Legendre Polynomials :   ( Back to Top )

InfProd_1109.gif

InfProd_1110.gif

InfProd_1111.gif

InfProd_1112.gif

InfProd_1113.gif

InfProd_1114.gif

InfProd_1115.gif

InfProd_1116.gif

InfProd_1117.gif

Series of Zeta, PolyGamma, PolyLog and related :   ( Back to Top )

InfProd_1118.gif

InfProd_1119.gif

InfProd_1120.gif

InfProd_1121.gif

InfProd_1122.gif

InfProd_1123.gif

InfProd_1124.gif

InfProd_1125.gif

InfProd_1126.gif

InfProd_1127.gif

InfProd_1128.gif

InfProd_1129.gif

InfProd_1130.gif

InfProd_1131.gif

InfProd_1132.gif

InfProd_1133.gif

InfProd_1134.gif

InfProd_1135.gif

InfProd_1136.gif

InfProd_1137.gif

InfProd_1138.gif

InfProd_1139.gif

InfProd_1140.gif

InfProd_1141.gif

InfProd_1142.gif

InfProd_1143.gif

InfProd_1144.gif

InfProd_1145.gif

InfProd_1146.gif

InfProd_1147.gif

InfProd_1148.gif

InfProd_1149.gif

InfProd_1150.gif

InfProd_1151.gif

InfProd_1152.gif

InfProd_1153.gif

InfProd_1154.gif

InfProd_1155.gif

InfProd_1156.gif

InfProd_1157.gif

InfProd_1158.gif

InfProd_1159.gif

InfProd_1160.gif

InfProd_1161.gif

InfProd_1162.gif

InfProd_1163.gif

InfProd_1164.gif

InfProd_1165.gif

InfProd_1166.gif

InfProd_1167.gif

InfProd_1168.gif

InfProd_1169.gif

InfProd_1170.gif

InfProd_1171.gif

InfProd_1172.gif

InfProd_1173.gif

Gauss Multiplication formula of Gamma functions, apply D[Log[#], z] on both sides m + 1 times gives

InfProd_1174.gif

InfProd_1175.gif

InfProd_1176.gif

InfProd_1177.gif

InfProd_1178.gif

InfProd_1179.gif

InfProd_1180.gif

InfProd_1181.gif

InfProd_1182.gif

InfProd_1183.gif

InfProd_1184.gif

InfProd_1185.gif

InfProd_1186.gif

InfProd_1187.gif

InfProd_1188.gif

InfProd_1189.gif

InfProd_1190.gif

InfProd_1191.gif

InfProd_1192.gif

InfProd_1193.gif

InfProd_1194.gif

InfProd_1195.gif

InfProd_1196.gif

InfProd_1197.gif

InfProd_1198.gif

InfProd_1199.gif

InfProd_1200.gif

InfProd_1201.gif

InfProd_1202.gif

InfProd_1203.gif

InfProd_1204.gif

InfProd_1205.gif

InfProd_1206.gif

InfProd_1207.gif

InfProd_1208.gif

InfProd_1209.gif

InfProd_1210.gif

Series of Beta Functions :   ( Back to Top )

Recurrence relation : Beta[x, a + 1, b] + Beta[x, a, b + 1] = Beta[x, a, b];

InfProd_1211.gif

InfProd_1212.gif

InfProd_1213.gif

InfProd_1214.gif

InfProd_1215.gif

InfProd_1216.gif

InfProd_1217.gif

InfProd_1218.gif

InfProd_1219.gif

InfProd_1220.gif

InfProd_1221.gif

InfProd_1222.gif

InfProd_1223.gif

InfProd_1224.gif

InfProd_1225.gif

InfProd_1226.gif

InfProd_1227.gif

InfProd_1228.gif

InfProd_1229.gif

InfProd_1230.gif

InfProd_1231.gif

InfProd_1232.gif

InfProd_1233.gif

InfProd_1234.gif

InfProd_1235.gif

InfProd_1236.gif

InfProd_1237.gif

InfProd_1238.gif

InfProd_1239.gif

Special values of InverseBetaRegularized :

InfProd_1240.gif

Series of Gamma Functions :   ( Back to Top )

InfProd_1241.gif

Dougall' s Formula :

InfProd_1242.gif

InfProd_1243.gif

InfProd_1244.gif

InfProd_1245.gif

( K[x] = EllipticK[x] ):

InfProd_1246.gif

InfProd_1247.gif

InfProd_1248.gif

InfProd_1249.gif

InfProd_1250.gif

InfProd_1251.gif

InfProd_1252.gif

InfProd_1253.gif

InfProd_1254.gif

InfProd_1255.gif

Special values of InverseGammaRegularized :

InfProd_1256.gif

Series involving HarmonicNumber : ( Back To Top )

InfProd_1257.gif

InfProd_1258.gif

InfProd_1259.gif

InfProd_1260.gif

InfProd_1261.gif

InfProd_1262.gif

InfProd_1263.gif

InfProd_1264.gif

InfProd_1265.gif

InfProd_1266.gif

InfProd_1267.gif

InfProd_1268.gif

InfProd_1269.gif

InfProd_1270.gif

InfProd_1271.gif

InfProd_1272.gif

InfProd_1273.gif

InfProd_1274.gif

InfProd_1275.gif

InfProd_1276.gif

InfProd_1277.gif

InfProd_1278.gif

InfProd_1279.gif

The notation in the Exponent of the hypergeometric function says : take the derivative with respect to the seventh argument and then give it the value 1 - a or 1 + a.

InfProd_1280.gif

InfProd_1281.gif

Series involving Hypergeometric Functions : ( Back to Top )

InfProd_1282.gif

InfProd_1283.gif

InfProd_1284.gif

InfProd_1285.gif

InfProd_1286.gif

InfProd_1287.gif

InfProd_1288.gif

InfProd_1289.gif

a few Integrals :   ( Back to Top )

InfProd_1290.gif

Substitute  InfProd_1291.gif   and the Feynman - Hibbs Integral

InfProd_1292.gif

InfProd_1293.gif

iterated expressions  ( Tetration ) :   ( Back to Top )

InfProd_1294.gif

InfProd_1295.gif

InfProd_1296.gif

InfProd_1297.gif

The above function f[x] = - ProductLog[-Log[x]] / Log[x] has a special 'swapping' symmetry of basis and exponent in its argument: InfProd_1298.gif.
f[x] is not defined beyond the maximum of its inverse function InfProd_1299.gif, namely  InfProd_1300.gif< x, so with this symmetry it is plausible that the exponential tower
doesn't converge for x < InfProd_1301.gif as well, where it shows a bifurcation.

some properties of ProductLog, LerchPhi and PolyLog   ( Back to Top )

For 1/e ≤ x    is ProductLog[ x Log[ x ]] =        Log[ x ] .
For 0 ≤ x ≤ e is ProductLog[ - Log[ x ] / x] = - Log[ x ] .
For 0 ≤ x       is Log[ ProductLog[ x ] ] =         Log[ x ] - ProductLog[ x ] .

InfProd_1302.gif

InfProd_1303.gif

For purely imaginary arguments (x ∈ R) the complex decomposition of LerchPhi is:

InfProd_1304.gif

These carry over with a = 0 to PolyLog:

InfProd_1305.gif

InfProd_1306.gif

The imaginary part of LerchPhi[x, s, a] with 1 ≤ x ∈ R is given by :

InfProd_1307.gif

And with a = 0 follows the imaginary part of PolyLog[ s, x] :

InfProd_1308.gif

The complex decomposition of  InfProd_1309.gif with 1 ≤ x ∈ R and 0 ≤ {b, s} ∈ N into real and imaginary part can be obtained by the following expression:

InfProd_1310.gif

explicitly for low s and b = 2 :

InfProd_1311.gif InfProd_1312.gif
InfProd_1313.gif InfProd_1314.gif
InfProd_1315.gif InfProd_1316.gif
InfProd_1317.gif InfProd_1318.gif
InfProd_1319.gif InfProd_1320.gif
InfProd_1321.gif InfProd_1322.gif

For all z ∈ C not on the real axis in ( -∞ < z < 1) and 0 ≤ {b, s} ∈ N the following inversion identity holds
(the If statement makes a '+' in case of an imaginary part of z larger than zero, a '-' in all other cases):

InfProd_1323.gif

The real part of  InfProd_1324.gifwith 1 ≤ x ∈ R is also given by

InfProd_1325.gif

For (b ∈ N) is

InfProd_1326.gif

The real and imaginary parts of LerchPhi[ InfProd_1327.gif, 2, 1/2 ] (on the unit circle) are

InfProd_1328.gif

With Clausen type functions for LerchPhi defined as

InfProd_1329.gif

InfProd_1330.gif

(0 < s ∈ Integer, 0 ≤ θ ≤ 2π, the even CLi and the odd SLi are expressible through Euler Polynomials),
the real and imaginary parts of InfProd_1331.gifInfProd_1332.gif (on the unit circle) are

InfProd_1333.gif

the expressions for InfProd_1334.gif with lowest s being

InfProd_1335.gif InfProd_1336.gif
InfProd_1337.gif InfProd_1338.gif
InfProd_1339.gif InfProd_1340.gif
InfProd_1341.gif InfProd_1342.gif
InfProd_1343.gif InfProd_1344.gif
InfProd_1345.gif InfProd_1346.gif

The above polynomials in a make nice approximations to trigonometric functions, getting better with increasing s.
The first non polynomial partnerfunctions are found to be

InfProd_1347.gif

The function InfProd_1348.gif has an interesting derivative :

InfProd_1349.gif

that means the lower CLi and SLi are essentially derivatives of the higher ones.

With the LerchPhi index n being a negative Integer the function appears as a rational function :

InfProd_1350.gif

With the PolyLog index being a negative Integer the function appears as a rational function :

InfProd_1351.gif

With Clausen type functions defined as

InfProd_1352.gif

InfProd_1353.gif

(0 < s ∈ Integer, 0 ≤ θ ≤ 2π,  the even Ci and the odd Si are expressible through Bernoulli Polynomials),
the real and imaginary parts of InfProd_1354.gif (on the unit circle) are

InfProd_1355.gif

the expressions for InfProd_1356.gif with lowest s being

InfProd_1357.gif

The above polynomials in a make nice approximations to trigonometric functions, getting better with increasing s:

InfProd_1358.gif

As before the derivative InfProd_1359.gifInfProd_1360.gifis InfProd_1361.gif with lowered index.
The first non polynomial partnerfunctions are found to be

InfProd_1362.gif

The complex decomposition of  PolyLog[s, x] with 1 ≤ x ∈ R and 0 ≤ s ∈ N can be obtained by the following expression:

InfProd_1363.gif

explicitly for low s :

InfProd_1364.gif InfProd_1365.gif
InfProd_1366.gif InfProd_1367.gif
InfProd_1368.gif InfProd_1369.gif
InfProd_1370.gif InfProd_1371.gif
InfProd_1372.gif InfProd_1373.gif
InfProd_1374.gif InfProd_1375.gif
InfProd_1376.gif InfProd_1377.gif

For all z ∈ C and not on the real axis in ( 0 ≤ z < 1 ) and 0 ≤ {b, s} ∈ N the following inversion identity holds :

InfProd_1378.gif

LerchPhi and PolyLog display a similar (alternating with s) scheme in their real and imaginary parts :

InfProd_1379.gif

InfProd_1380.gif

InfProd_1381.gif

InfProd_1382.gif

InfProd_1383.gif

InfProd_1384.gif

InfProd_1385.gif

InfProd_1386.gif

The lowest Bernoulli and Euler Polynomials are

BernoulliB EulerE
InfProd_1387.gif InfProd_1388.gif InfProd_1389.gif
InfProd_1390.gif InfProd_1391.gif InfProd_1392.gif
InfProd_1393.gif InfProd_1394.gif InfProd_1395.gif
InfProd_1396.gif InfProd_1397.gif InfProd_1398.gif
InfProd_1399.gif InfProd_1400.gif InfProd_1401.gif
InfProd_1402.gif InfProd_1403.gif InfProd_1404.gif

They are symmetric or antisymmetric (depending on n) with respect to x = 1/2 :

InfProd_1405.gif

Clausens Integral :

InfProd_1406.gif

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