the generalization of special functions

Applied Mathematics and Computation 187 (2007) 395–402

www.elsevier.com/locate/amc

The generalization of special functions

Asghar Qadir

Centre for Advanced Mathematics and Physics, National University of Sciences and Technology,

Campus of the College of Electrical and Mechanical Engineering, Peshawar Road, Rawalpindi, Pakistan

Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

Dedicated to Professor H.M. Srivastava on the occasion of his 65th birthday

Abstract

The question of what special functions should be taken to be is addressed. Using examples of generalizations that haveproved useful, with special reference to those developed by Chaudhry and me, an attempt is made to formulate criteria forguiding us in the search for worth while generalizations of special functions.� 2006 Elsevier Inc. All rights reserved.

Keywords: Extension; Generalization; Special functions

1. Introduction

There is no real definition of which functions are to be regarded as ‘‘special functions’’. Yet there are booksand courses on them and they are heavily used in analytical number theory, engineering mathematics and inmathematical physics. Whatever they may be, it seems contradictory to talk of generalizing them. If they arespecial they are not general and vice versa. Nevertheless, there has been a lot of work done on their general-ization, and those generalizations have been heavily used. It is worth while to step back and consider what thisenterprize of generalization is and what it amounts to. In particular, there can be infinitely many generaliza-tions of any function that introduce extra parameters and extend the domain of the generalized function (or insome cases reduce it from infinity). Most of them would appear to be futile but some may lead to new insightsand have far-reaching applications. How can we know that we are following up a useful generalization or, onthe other hand, not throwing away a valuable generalization?

Let us first address the question of what special functions are. They have been said to be those functionsthat appear at least once in the literature. They have often been regarded as those functions that arise in thesolution of the second order linear partial differential equations arising in physical and engineering problems[15]. They have been taken to be intuitively apprehended to be special. There are other attempts to specifythem. However, none of these views regarding special functions is problem-free. If the first view is taken, then

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396 A. Qadir / Applied Mathematics and Computation 187 (2007) 395–402

every known function is special and hence there are no non-special functions. That cannot begin to makesense. The second view excludes the gamma, beta (see for example [8]) and zeta functions [16] which mustsurely be regarded as special. The third view makes the definition subjective.

Perhaps the easiest solution of the problem of definition would be to list the special functions. By itself, thiswould also be inadequate as it limits the functions to be regarded as special. Why should new functions beregarded as special? Before the hypergeometric functions [12] were defined there were other special functions.Had the list been taken as limited to those defined before the hypergeometric function, the latter would havebeen excluded. However, from our present point of view they would be regarded as special. More generally, ifnew functions give previously known special functions, that may make them special. This brings into focus thequestion ‘‘what is meant by some functions being related to other functions?’’ Further, why should ‘‘relatedfunctions’’ be of interest?

Here the issue of what special functions are will be discussed, with special reference to the significance of thegeneralization of special functions. To attempt to answer these questions I shall present some generalizationsof various special functions that Chaudhry and I presented and try to draw lessons from them regarding whatsort of generalization may be regarded as useful, and why. Finally, some concluding remarks will be made.

2. The list of special functions

The term ‘‘special functions’’ is applied to functions that are not trivial. In some sense the polynomials arethe most ‘‘special’’ of all to the extent of being ‘‘boring’’. The next simplest is the family of exponential func-tions the usual real exponential, the trigonometric and hyperbolic functions, the general complex exponentialthat incorporates all of them and their inverse functions. These may have been supposed to be ‘‘boring’’ but,as we shall see, they are interesting. They can be taken as solutions of first order differential equations.

The next lot of special functions is the solutions of the linear second order differential equations of math-ematical physics and engineering. These include the Legendre polynomials and functions, the Laguerre, theHermite, the Bessel and related functions and some others. All of them arise as special cases of the hyper-geometric, or confluent hypergeometric, functions [12] that are solutions of the hypergeometric equation

zð1� zÞw00 þ ½c� ðaþ bþ 1Þz�w0 � abw ¼ 0: ð1Þ
Notice that the generalization of the other special functions has proved even more useful than the separatespecial functions themselves.
The other functions are those that arose in the theory of numbers. It is of interest to follow them in moredetail. The first of them, the gamma function [8], generalized the factorial function defined for integers to thereal domain and thence to the complex domain

CðaÞ :¼Z 1

0

e�tta�1 dt: ð2Þ

Notice that the generalization extended the domain of the original function. This is one of the principal benefitsof the generalization of special functions. Unlike the hypergeometric functions, which fulfilled the purpose theywere constructed for, namely as solutions of those second order equations that arose repeatedly, the gammafunction was ubiquitous turning up in all sorts of problems. The same applies to the beta function, which arisesfrom the gamma but is used in areas like statistics and probability, and to Riemann’s zeta function (and theother members of the zeta function family). Again note the appearance of the generalizations of functions.

There is a view that with the advent of computing power the study of special functions has become redun-dant. The attitude seems to be that functions are simply lists of outputs for given inputs and hence we onlyneed solve the problems numerically. This view ignores the fact that numerical solutions can be unreliablewithout mathematical analysis to validate them and it ignores the power and the insights that are obtainedfrom analytical solutions where numerical solutions may fail. Holders of that view are particularly criticalof the generalizations that are put forward for the special functions. Here there is a valid point. There are infi-nitely many generalizations possible. In fact, since there are @2 possible functions, there are as many possiblegeneralizations. If we require that the functions be analytic or C1 or Cn (for some natural number n), there arestill @1 possible generalizations for every special function. Consequently, we need some criteria that will limit

A. Qadir / Applied Mathematics and Computation 187 (2007) 395–402 397

the number of generalizations to those that are significant. The essence of the criteria is present in the earliersuccessful generalizations. Let us examine those in a bit more detail.

3. Generalizations of the special functions of analytical number theory

The generalization may give the previously known special functions as special cases. How is this useful?Among other insights, it provides a connection, or new connections that had not been seen before, betweenseemingly unrelated functions. And how does that help? Again, among other things, it can provide freshinsights into the function that was being generalized and thus provide properties, or proofs, that were lackingbefore. I illustrate with some examples that I have been involved in.

The gamma function can be broken into the two incomplete gamma functions [8] by breaking the range ofintegration for Eq. (2) at some finite positive number x. The incomplete gamma functions are generalizationsof the original gamma function by restricting the domain of integration. They prove useful in problems thatare formulated over finite, instead of semi-infinite domains. In certain situations they did not solve the prob-lem but a generalization

cbða; xÞ :¼Z x

0

e�t�b=tta�1 dt; ð3Þ

(and the corresponding complementary function Cb(a,x)) did [9]. It was seen to be useful in many other prob-lems as well [8].

The zeta function [10], and especially the Riemann hypothesis, is important in the theory of prime numbers[11]. To try to throw fresh light on the Riemann hypothesis, the above type of generalization was attemptedfor the zeta function [4], namely

fbðaÞ :¼Z 1

0

ta�1ð1� e�tÞ�1e�t�b=t dt; ð4Þ

with Re(b) > 0. It was found that this extended zeta function could be expressed as a series expansion of thegeneralized gamma function and of the Macdonald function. These connections provided strong reason forregarding the generalization as useful in its own right. It is strongly damped near t = 0 due to the e�b/t andhence passes through the value r = 1. Thus the extension provides a ‘‘regularization’’ of the zeta function.Of course, in the ‘‘critical strip’’ for 0 < r < 1 the behaviour of the extended function will not match thatof the original zeta function. It was hoped that this difference would yield results on the behaviour of the zetafunction in the critical strip by relating to another extension

f�bðaÞ :¼ ½CðaÞð1� 21�aÞ��1

Z 1

0

ta�1ð1þ etÞ�1e�b=t dt: ð5Þ

The relation between them is

f�bðaÞ ¼ ½f2bðaÞ � 21�afbðaÞ�=ð1� 21�aÞ: ð6Þ
The hope (of proving the Riemann hypothesis) was never justified but the relation provided various inequal-ities between the two extensions and the second extension could be written as a series of Bernoulli numbers[17]. The extensions also led to an extension of the Hurwitz formula [16].
Inequalities of the Riemann zeta function are very important [13,14] and there are various bounds knownfor it. Though the above extensions were not, themselves, involved in providing sharper bounds for the ori-ginal zeta functions, the relations obtained between the first and second extensions and the Bernoulli numbers(and hence Bernoulli polynomials) did provide the means to obtain sharper bounds [1] for f(x + 1) in terms off(x)

½1� bðxÞ�fðxÞ þ bðxÞ86 fðxþ 1Þ 6 ½1� bðxÞ�fðxÞ þ bðxÞ

2; ð7Þ

where

bðxÞ :¼ ð2x � 1Þ�1: ð8Þ


This inequality leads to the estimate for f(x + 1)

fðxþ 1Þ ¼ ½1� bðxÞ�fðxÞ þ 5

16bðxÞ þ EðxÞ; ð9Þ

where

jEðxÞj < e ð10Þ
for
x > x� :¼ ln 2 � ln 1þ 3

16e

� �: ð11Þ

Various other useful bounds were also obtained.The same procedure of extension was not available for the beta function, because of the fact that it is

defined over a finite domain of the parameter, thereby providing symmetry in the two variables on which itdepends

Bðx; yÞ ¼Z 1

0

tx�1ð1� tÞy�1 dt; ð12Þ

where x and y are complex numbers with positive real values. The factor e�b/t would destroy that symmetry.To retain the symmetry we used the extension [5]

Bðx; y; bÞ :¼Z 1

0

tx�1ð1� tÞy�1e�b=½tð1�tÞ� dt; ð13Þ

with Re(b) > 0. It is clear that the earlier restriction on the real parts of the arguments no longer applies,because of the exponential damping of the extended function. This extension has very interesting relationsto the original beta function, namely
Z 1
0

bs�1Bðx; y; bÞdb ¼ CðsÞBðxþ s; y þ sÞ; ð14Þ

with the requirement that Re(x + s), Re(y + s) > 0. Using negative integer values of s gives a formula for thenth derivative relative to b, with the C replaced by a factor of (�1)s. Of particular interest is the case of s = 1,which relates the integral of the extended beta function over the new parameter, to the shifted beta function,with the domain extended to Re(x), Re(y) > �1.

A special case of the function turned out to be directly related to the Macdonald function

Bða;�a; bÞ ¼ 2e�2bKað2bÞ ð15Þ
and the Whittaker function
Bða; a; bÞ ¼ pp2�abða�1Þ=2W �a=2;a=2ð4bÞ: ð16Þ

The extended beta function shifted by a negative integer can be replaced by a finite series expression of Mac-donald functions and by a positive integer of Whittaker functions. Conversely, the Macdonald and Whittakerfunctions can be expressed as finite series expansions of the extended beta function. It also turns out to beuseful in the theory of reliability. The relations of the extended function to the original function, to otherknown functions, the extension of the domain and the utility in problems for which it was not constructed,make it a useful generalization.

4. Generalizations of special functions that are solutions of the partial differential equations of mathematical

physics

The special functions of mathematical physics arise from solution of differential equations that arise inphysics. The simplest such equation is the first order equation that gives the exponential function. Asmentioned earlier, that may seem too trivial to give any interesting generalization but in fact it does give an


interesting extension. The idea is to construct an incomplete version of it, not by truncating or breaking intothe even and odd parts, but by using the incomplete gamma function [2]

eððx; tÞ; aÞ :¼X1n¼0

cðaþ n; xÞCðaþ nÞ

tn

n!; ð17Þ

Eððx; tÞ; aÞ :¼X1n¼0

Cðaþ n; xÞCðaþ nÞ

tn

n!; ð18Þ

so that

eððx; tÞ; aÞ þ Eððx; tÞ; aÞ ¼ et: ð19Þ
This has the elegant extension of the derivative property of the usual exponential as
o

oteððx; tÞ; aÞ ¼ eððx; tÞ; aþ 1Þ ð20Þ

and a similar expression for E((x, t);a). This use shows the benefit of defining the new function. They also finda direct application in Statistics.

The incomplete exponential functions are related to the hypergeometric function 0F1(�;a; tx) by

o

oxeððx; tÞ; aÞ ¼ xa�1e�x

CðaÞ 0F 1ð�; aþ 1; txÞ; ð21Þ

o

oxEððx; tÞ; aÞ ¼ � xa�1e�x

CðaÞ 0F 1ð�; aþ 1; txÞ; ð22Þ

so that the derivative of the sum of the two is independent of x. This leads to the result

o

ot0F 1ð�; a; txÞ ¼ o

ox0F 1ð�; aþ 1; txÞ: ð23Þ

These incomplete exponentials can be extended to provide an incomplete extension for the generalized hyper-geometric functions pFq.

The ‘‘incomplete’’ generalization is not the only extension of the hypergeometric functions possible. Wecould use the extension procedure adopted for the beta functions on account of the fact that the hypergeomet-ric and confluent functions can be expressed as power series with the beta functions as coefficients. Thus, since

2F 1ða; b; c; zÞ ¼ CðcÞCðbÞCðc� bÞ

X1n¼0

ðaÞnBðbþ n; c� bÞ zn

n!; ð24Þ

1F 1ðb; c; zÞ ¼ CðcÞCðbÞCðc� bÞ

X1n¼0

Bðbþ n; c� bÞ zn

n!; ð25Þ

where

ðaÞn ¼Cðaþ nÞ

CðaÞ ð26Þ

and the real parts of b and c are positive, writing the hypergeometric function as F(a,b;c;z), we can define [6]

F pða; b; c; zÞ ¼X1n¼0

Bðbþ n; c� b; pÞBðb; c� bÞ ðaÞn

zn

n!; ð27Þ

with p P 0. Similarly, writing the confluent hypergeometric function as U(b;c;z) we have

Upðb; c; zÞ ¼X1n¼0

Bðbþ n; c� b; pÞBðb; c� bÞ

zn

n!: ð28Þ

All the usual properties of the hypergeometric and confluent hypergeometric functions carry over directly fortheir extensions. The connection of the extended beta function with the Macdonald and Whittaker functionspointed out above, lead to new relations of the hypergeometric function with these functions.


5. Generalizations of other special functions

The integral of the Gaussian function may be thought of as a ‘‘constant function’’ whose ‘‘incompleteparts’’ are the error function, erf(x) and its complement erfc(x). Since the Gaussian function gives the normalprobability distribution, the error function gives the cumulative probability hence the name. It arises not onlyin statistical analyses but also in the solution of differential equations. The error functions can be written interms of the incomplete gamma functions as

erfðxÞ ¼ 1ffiffiffipp cð1=2; x2Þ ð29Þ

and

erfcðxÞ ¼ 1ffiffiffipp Cð1=2; x2Þ: ð30Þ

Consequently, the generalization of the gamma and incomplete gamma functions can be used [7] to generalizethe error functions to give

erfðx; bÞ ¼ 1ffiffiffipp cð1=2; x2; bÞ ð31Þ

and

erfcðx; bÞ ¼ 1ffiffiffipp Cð1=2; x2; bÞ: ð32Þ

This generalization is useful in solving problems of heat conduction and of reliability.There can be ‘‘spin-offs’’ of the generalization procedure in that the same procedures could be applied to

other functions to produce useful results. The inequalities for the zeta functions was a case in point. Asanother example, the extension and generalization procedures adopted relied on the integral representationsof the functions and the use of transforms for them. By using the Fourier transform it was possible to write thegamma function as a series of Dirac delta functions, thereby providing a distributional representation of thegamma function [3]. This representation should prove useful in solving boundary value problems where agamma function source term appears. In any case, it leads to new identities for integrals of the gamma func-tion. For example
Z �1
�1jCðrþ isÞCðqþ isÞj2 ds ¼ 2pC2ðqþ rÞBð2r; 2qÞ ð33Þ

for positive q, r. In particular, taking q = r gives
Z �1
�1jCðrþ isÞj4 ds ¼ 2pC2ð2rÞ=Cð4rÞ: ð34Þ

Another example is

hCðrþ isÞ;KrþisðxÞi ¼ 2pX1n¼0

ð�1Þn

n!KnðxÞ: ð35Þ

6. Conclusion

I have presented various examples of generalizations of special functions that have been applied to deep-ening an understanding of the functions themselves and to engineering and statistical problems. It is timeto return to the question addressed at the start of this paper.

The first question was ‘‘what are special functions?’’ As we have seen, it would be appropriate to call thosefunctions ‘‘special’’ that occur repeatedly and are used in deepening the understanding of functions or in


application of mathematics in other areas, such as Physics, Engineering, Statistics, etc. Their utility is whatmakes them ‘‘special’’.

The next question was ‘‘why generalize them’’? After all, if they are special they are not general and viceversa. The point is that the same criterion of what makes functions special guides us in generalizing them.If the generalization, or extension, is useful in any of the above-mentioned ways then it is a ‘‘good’’generalization.

Finally, we must address the question of ‘‘what are the uses that we should look for?’’ This question is mostimportant, as there are increasingly many new functions ‘‘appearing on the market’’ and we need ‘‘consumerprotection’’ from those that are not ‘‘good’’. From the above examples I shall try to extract the essential cri-teria for ‘‘good generalizations’’. All the criteria may, or may not, be met. However, at least some should bemet. The more that are met, the ‘‘better’’ the generalization.

1. The new function carries over most of the properties of the original function with ‘‘minor’’ modifications toallow for the difference.

2. It provides insights into the original function that lead to advances in our understanding of that function(probably by providing a fresh angle of viewing it).

3. It leads to new results for the original function.4. It changes the domain or range of the original function, probably by passing over a point or region of sin-

gularity, so that it can be applied to problems that were inaccessible earlier.5. It provides relations between different functions, generalizing them into special cases of a single ‘‘special

function’’.6. It recurs in the solution of different problems the more wide-ranging the problems, the better the

generalization.7. It solves problems other than the one it was constructed for.

It must be admitted that not all the points mentioned above are totally independent of each other. How-ever, it is better, in this case, to err on the side of redundancy than to omit one of the criteria that would makethe function ‘‘worth while’’.

Acknowledgements

The author is grateful to the University of Victoria and the Pacific Institute of Mathematical Sciencesfor the invitation to present this talk at the International Dedication Symposium in honour of the 65th birth-day of Prof. H.M. Srivastava.

References

[1] P. Cerone, M.A. Chaudhry, G. Korvin, A. Qadir, New inequalities involving the zeta function, J. Inequal. Pure Appl. Math. 5 (2004)1–17. Article 43.

[2] M.A. Chaudhry, A. Qadir, Incomplete exponential and hypergeometric functions with applications to the non-central v2 distribution,Commun. Statist. Theory Meth. 34 (2005) 525–535.

[3] M.A. Chaudhry, A. Qadir, Fourier transforms and distributional representations of the gamma function, leading to some newidentities, Internat. J. Math. Math. Sci. 37 (2004) 2091–2096.

[4] M.A. Chaudhry, A. Qadir, M.T. Boudjelkha, M. Rafique, M.S. Zubair, Extended Riemann zeta functions, Rocky Mountain J. Math.31 (2001) 1237–1263.

[5] M.A. Chaudhry, A. Qadir, M. Rafique, M.S. Zubair, Extension of Euler’s beta function, J. Comput. Appl. Math. 78 (1997) 19–32.[6] M.A. Chaudhry, A. Qadir, H.M. Srivastava, R.B. Paris, Extended hypergeometric and confluent hypergeometric functions, Appl.

Math. Comput. 159 (2004) 589–602.[7] M.A. Chaudhry, A. Qadir, M.S. Zubair, Generalized error functions with applications to probability and heat conduction, Internat.

J. Appl. Math. 9 (2002) 259–278.[8] M.A. Chaudhry, S.M. Zubair, On a Class of Incomplete Gamma Functions with Applications, Chapman and Hall (CRC Press),

London, 2001.[9] M.A. Chaudhry, S.M. Zubair, Generalized incomplete gamma functions with applications, J. Comput. Appl. Math. 55 (1994) 99–124.

[10] H.M. Edwards, Riemann’s Zeta Function, Academic Press, New York, 1974.


[11] G.H. Hardy, J.E. Littlewood, Contributions to the theory of the Riemann zeta functions and the theory of the distribution of primes,Acta Math. 41 (1918) 119–196.

[12] L.J. Slater, Generalized Hypergeometric Functions, Cambridge University Press, Cambridge, 1966.[13] H.M. Srivastava, Some rapidly converging series for f(2n + 1), Proc. Amer. Math. Soc. 127 (1999) 385–396.[14] H.M. Srivastava, Some families of rapidly convergent series representations for the Zeta functions, Taiwanese J. Math. 4 (200) 569–

596.[15] N.M. Temme, Special Functions: An Introduction to the Classical Functions of Mathematical Physics, Wiley, New York, 1996.[16] E.C. Titchmarsh, The Theory of the Riemann Zeta-Function, Oxford University Press, Oxford, 1951.[17] E.T. Whittaker, G.N. Watson, A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of

Analytic Functions; with an Account of the Principal Transcendental Functions, fourth ed., Cambridge University Press, Cambridge,1927.

the generalization of special functions

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