SOME THEOREMS ON THE EXPLICIT EVALUATION OF RAMANUJAN ’ S THETA-FUNCTIONS NAYANDEEP DEKA BARUAH and

Bruce C. Berndt et al. and Soon-Yi Kang have proved many of Ramanujan’s formulas for the explicit evaluation of the Rogers-Ramanujan continued fraction and theta-functions in terms of Weber-Ramanujan class invariants. In this note, we give alternative proofs of some of these identities of theta-functions recorded by Ramanujan in his notebooks and deduce some formulas for the explicit evaluation of his theta-functions in terms of WeberRamanujan class invariants. 2000 Mathematics Subject Classification: 11A55, 11F20, 11F27.

At scattered places in his notebooks [8], Ramanujan recorded several values of his theta-function φ(q).Proofs of all the values claimed by Ramanujan can be found in [3,Chapter 35].Berndt and Chan [4] also verified all of Ramanujan's nonelementary values of φ(e −nπ ) and found three new values for n = 13, 27, and 63.Kang [6] also calculated some quotients of theta-functions φ and ψ.In Section 3, we give some theorems for the explicit evaluation of the quotients of theta-functions φ, ψ, and f , by combining Weber-Ramanujan class invariants with the identities proved in Section 2 and some other identities of theta-functions.Some of these evaluations can be used to find explicit values of the famous Rogers-Ramanujan continued fraction R(q) defined by where |q| < 1.

Theta-function identities.
The following identity was recorded by Ramanujan on page 295 of his first notebook [8].Berndt [3,page 366] proved this by using parameterization.Here we give an alternative proof.
Since G 9n and g 9n can be calculated from the respective values of G n and g n [5], from the theorem above, we see that the quotients of theta-functions on the left-hand sides can be evaluated if the corresponding values of G n and g n are known.We give a few examples below.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.
However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation