The function shifts the characters in the input string forward in the alphabet based on their position (index+1), while maintaining their case and leaving numbers and special characters unchanged.

[CODE]: 
```python
def approximate_function(input_string):
    output = ""
    for index, char in enumerate(input_string):
        if char.isalpha():
            shift = (index + 1) % 26
            if char.isupper():
                new_char = chr(((ord(char) - ord("A") + shift) % 26) + ord("A"))
            else:
                new_char = chr(((ord(char) - ord("a") + shift) % 26) + ord("a"))
            output += new_char
        else:
            output += char
    return output
```

Please test the function with more inputs if you would like to confirm or refine this description further.